What is my sister thinking / feeling, and how do I deal?

The question, really, is does someone have POA for healthcare? If it's your sister, that gives the rights to call the shots, even if you are sitting there.

Find out.

Find out who had POA for healthcare. Realize that your father can assign a new one if he is mentally competent to show he understands what it means.

Try to bury all this emotional stuff with your sister. This isn't about her or about you. It is about your father. What does he want?

sounds like counseling would benefit both of you, is the center where dad is have a family support group sessions.? Maybe you can talk with facilitator and she can help you and sister navigate this tricky relationship and responsibility roles. He or she may have some good ideas or even be willing to help you both compromise for sake of dad.

Sibling relationships are hard. If she has been primary caregiver for some time, and you have been disengaged, she may resent your riding in and taking over. She isn’t seeing it as help but an interference.

Seek help from others by joining local caregiver support group at dads facility. You’ll learn a lot.

Thank you for the comments. I was remembering that we both had POA, both medical and durable. However, I am only listed as secondary on the medical documents. We have both been taking care of him for the past five years. I have done most of the in-person "heavy lifting", and she has taken care of the administrative stuff.
What really set me off is that he was recently in the hospital due to pain from a fall, and delirium. He was deemed over-medicated and said that probably contributed to the fall. The SNF Dr had recently doubled his anti-depressant and added Buspar, plus his Exelon patches were not being removed and that was building up in his system. The Dr in the hospital discussed that with both of us and suggested the meds be cut back, which we agreed to. He started to improve and was transferred back to the SNF. I trusted my sister, as I always have, to make sure all was in order when he transferred back to the SNF (I was out of town taking care of my uncle). Since he has been back he has slipped into delirium again - even our caregiver remarked that he's just not himself.
I have trusted her, and think I still do, to do what is best, but she has a habit of resisting in any area, no matter how petty, where I have a bit more experience. Depression is not petty, and is something I share with my dad that she knows little about, luckily for her. A few years ago she unilaterally tried to take him off anti-depressants and, though I was annoyed when I found out, I was not too mad, and made it clear that I wanted to be consulted in any psych issues with him.
Back to a few days ago, she says "Oh, well he is still on the higher doses of psych meds - they weren't changed on the discharge paperwork" She said a few more things that led me to believe she'd known this for some time, and that she didn't tell me because "things seemed to be going OK."
I guess she decided to tell me right before the care meeting, in case it came up - she knew I'd be surprised. So after the care meeting I texted her about which one of us was going to deal with the staff, and said I was uncomfortable and wanted to discuss. She texted back "Who is handling what is a topic not for text and for another day". That was what sent me through the roof - her tone and dismissal of me.
We had a big blowup last summer over similar matters, and I did set up a counseling session for both of us to attend on her next visit. She did not want to go due to her work, and though I tried and tried to say "let's sit down and talk when you are here" it doesn't happen.
Not surprisingly we've had this issue and pattern with finances too - I try to talk with her about it and she avoids it - kicks the can down the road.
There have been some times that we worked really well together, like sisters - but it feels like she is acting more like my dad'd wife and I'm the child. Not surprisingly my dad's ex-wives also felt this way about her relationship with him.
Well, writing this has helped. I see that I am doing a lot to try to have a sisterly relationship with her, and she is shutting me out. I guess she holds all the cards legally, so there is nothing I can do. Did I mention that I am the oldest?

The NH will discuss care issues with you, meaning you are on your dad’s list of whom they can share information with? It sounds like you are, so that’s a good thing.
It does appear that you are both involved in managing your dad’s care. One of you has to be the designated POA and your father assigned that role to your sister.
I don’t understand the real issue here. Do you want more control over your dad in what way?
Do you call your sister at a time when it’s convenient for her to talk? I don’t think it was inappropriate for your sister to tell you the matter shouldn’t be addressed by text and left for a better time. Does your sister work? Have a family? If so consider that she is managing a lot on her end.
You are there day to day- boots on the ground so to speak. Your father is receiving care in a facility and both of you appear to agree that the care being delivered is more than adequate. What more do you want from her?
Ask to set a time for you two to discuss your dad on a weekly or every two week basis. Whether it be a weekend or evening, whatever works for both of you.
Why is dad “missing out”, as you say Because you don’t have the relationship with your sister that you feel is adequate? It sounds like you both do care for him very well if both of you attend the care conferences.
Maybe if you look inward and determine what your expectations of your sister are, you can then talk about this with her. You both appear to be united in caring for your father so that’s a big plus. I am not sure what more you want, and maybe neither does your sister know what you expect from her.
Your last statement- that you are the oldest- leads me to think you feel the #1 POA should be you. Might that be true in that you feel you should be #1 & your younger sister the back up (as you see yourself?).
If so unless your sister is mismanaging dad’s care then let it go. Why stir the pot? Dad is fine. Sister continues to do the right thing. It sounds like you feel left out in some way, when in fact both of you are on the same page.
Choose your battles. You can’t read your sister’s mind so you will need to work on building the relationship between the both of you. That’s how you deal. 
Good luck to you all.

You moved back knowing you were listed as backup on the Healthcare POA. You need to respect that boundary - you had no reason to think it would change, and you've had issues with sis not doing what you wanted before. Legally, she can only resign the Healthcare POA, and then she can't be placed back on if you ditch. I would do exactly what she's doing: Following the POA designation.

Don't take this the wrong way. I too had a situation where the unpalatable reality was that my sister didn't trust me - me! How dare she! - as far as she could throw me. I felt indignant, I felt it was unreasonable, I felt it said a lot more about her than it did about me... But there it was. She didn't trust me. Unflattering, but a fact.

So where you say... "Well, writing this has helped. I see that I am doing a lot to try to have a sisterly relationship with her, and she is shutting me out."

... I wonder if writing how you think she might see things would help, too. What's she afraid of? Why might she resist the idea, for example, of resigning her health care responsibilities and allowing you to step up?

It's a question of seeing things "as they are, not as they ought to be." At least then you know what you've got to work round.

Thanks all. To get one thing straight, I'm not asking her to resign her POA to me. I'm asking her to make me a true partner in his care rather than delegating to me.

Shane, we are not trustful of the facility or any facility, and have learned that we need to be vigilant in advocating for him e.g. keeping on top of the staff. She does have a lot on her plate, as she has a job and I don't right now. But it's not really about that, I'm seeing.

Countrymouse, I think you are right on that she doesn't trust me, or, at least, she does not trust me to care and love her dad as much as she does. Yes, not our dad, but her dad. Since she was a young kid she has taken responsibility for his happiness - rather - he put her in that role when my mother divorced him. It's called parentification and is not healthy - it is sometimes referred to as emotional incest. She doesn't see it and maybe never will.

I could write a book on it all, but I'll spare you all here. That is the way things are, and I've gotten emotionally sucked back into the dysfunction and the way I want them to be. Now that I realize what has happened, I can deal with it better. This discussion has really helped me, so thank you.

I guess I should say too that I am no longer as distraught and angry as I was initially. She does have legal control, and that was my dad's choice. So it is now her choice on how much she will allow me to do, and I have asked her to let me know her thoughts. I also want to make sure she is not expecting me to do certain things that she and I have not discussed (and I have not agreed to) - because there is some of that going on too. We'll see what she says.

Twincarer, your sister has likely always been the dominant sibling and that isn't going to change now. No good deed goes unpunished, and even though you have moved to be near your dad, she's not going to suddenly say "sis, you're right, let's do it your way" at this point in life.

You can't choose her attitude. You can talk with the SNF staff. I'm wondering if you are on the list of people with access to your father's medical information? (That's a HIPPA issue, not a POA issue.) If he's legally competent to list you as someone who gets access to his info, then make sure the SNF knows that. If he hasn't listed you, get him to do so.

Do what you can, but just know the family dynamic is not going to change.

Emotional incest? Parentification? That sounds creepy. Do all of us caring for our parents have this? Can’t she just simply just care about her father?
Obviously their relationship was close if he made her POA. Maybe she is simply a daddy’s girl?
Sounds like you have put a lot of thought into figuring the hierarchy of their relationship so you can cope with it the best that you can. 
Does sister come visit often? She could be feeling guilty she doesn’t see him often and overcompensates by keeping a tight rein on him from afar?
Family dynamics are complex. I would just try to continue to communicate as best you can. At least you two are on the same page and care about your dad a lot.

Shane - yes, it is creepy. No, it's not about us caring for our parents, it's about emotionally needy parents co-opting their kids to make them buddies, best friends, dates, even. It gets brushed off as "cute" until people who are close to them realize that neither father or daughter have much of a life outside of each other. Now that we are grown, he's co-opted us both as his caretakers, though he has been able to take care of himself, if he wanted to. He has taken the last five years of our lives - she has gotten divorced and neither of us have the healthy lives we did before his last wife left him.

Oh yes, family dynamics are really complex, and yes, I have spent a lot of time thinking about how I can do my best to live within our tribe. I get it figured it out, then slip back into the pattern, then have a meltdown (see above), and then I get back on track. Yes, I love my dad and sister, despite the dynamics. I don't expect my dad to be aware of how he could have done differently by us.

A lot of my pain comes from my sister not being aware of the dynamics e.g. dysfunction. I want her to be my sister - I need her to be my sister. And yes, we do love all love each other.