Oh my Wend A.
First let me express how sorry I am that your dad is nearing the end of his life. That is so hard. Adding BPD mom into the mix just amplifies everything. I am happy you manage to see your dad fortnightly. I'm guessing that when you go, somehow your mom makes it about her with you around. That's just a guess. Hopefully it's off base.
Following my friendly “how are you” text, my BPD mum demanded I call. I explained I couldn’t make calls whilst working, had just snook a message but would call after my shift. She replied “it only takes a minute, forget it. Even if you can call, just don’t b**** bother”.
I have a suggestion. Boundaries

What I mean by that, is when you are at work, your boundary is that you don't answer her texts or emails. It's a boundary. This does not have to be announced to her (as it would probably create conflict). Just follow your own boundary, so next time when you see a text come in from her, don't answer it if you are at work. Answer it at a convenient time after your work is finished. Resist responding right away.
If she asks you what took so long, you can answer that you were at work. Leave it at that.
The problem here could be that you answered but
explained (JADE) that you couldn't make calls at work. Don't explain as this leaves her feeling invalidated, but also makes you look guilty because you are needing to explain yourself. Instead, just answer her text when you are ready. That way she will learn the boundary through natural consequences. Maybe if there is a calm time with her, and the right moment presents itself, you can sneak in that you can't answer personal texts or phone calls at work (maybe even in the context of not answering your
friends so she doesn't take it personally. Will she like this boundary? No. At first there may be an extinction burst, but she
will learn it, if you enforce it by consistently not answering when you are at work. If you answer her once while at work, she will expect you to answer again during work, and you will have to start the whole process over again.
You can make sure the home care facility has your contact information so that if your dad takes a turn, they can contact you directly. It's up to you if you want to let the facility know that you can't answer calls from your mom while you are at work. They may not be aware of the dynamic between the two of you.
I'm assuming you are probably familiar with JADE, but in case you aren't, here's a link:
https://bpdfamily.com/message_board/index.php?topic=139972.20Wend A, I understand how bad it feels (worst daughter on the planet as she is hurting and lonely). You are
not the worst daughter on the planet. I get that prize (according to my uBPD mother). In fact everyone on this board gets the same prize from their parent. Turns out there's a lot of "worst son/daughter's ever on the planet" cs there's a lot of BPD parents.
I empathize. But we are not evil. We are not mean. We are not horrible people. And we are not the worst daughter's ever. In fact, I suspect we are some of the best son's/daughters on the planet, because despite their toxicity, we keep trying, because they are our parent. I have done way more for my mom than most of my friends (good people), who have been stellar with their aging parents. Yet "I too am the worst daughter" ever. Something is wrong with that picture right?
So despite the awful feeling your mother somehow thrusts on you, ask yourself "what is the
evidence I am the worst daughter ever, according to normal standards (rather than your mother's standard)? Her words and thinking does not count as evidence.
Also, what are the things you have done that have been supportive to her and your dad through all this? If you took the time to write a list (every little detail), I'm guessing your list would be long. Thinking about this is not for her benefit. Telling her would be a waste of time and only make it worse. The benefit is for YOU, so that you can know and feel and remember that you are doing your best. She can feel differently. Those are her feelings. You can have your own feelings, which are different from hers. Having different feelings is normal. But we have not been raised to believe that.
As to your last question, you feel that way because your mother makes you feel that way to make herself feel better. It's the disease.