Take the opportunity to work
It is, essentially, being handed to you on a silver platter. So take it.
In the meantime, though, a few things.
1) You'll need to stay in the home at least for a while as you find a job and also get yourself on your feet financially. Job first, then apartment. It might be a year or so of steady working before you can really start actively looking for an apartment, so you should be set in terms of being able to be around.
2) While you're around, you're right, learn what you need to learn about caring for your mother. You are well aware of the financial situation so it's entirely possible that funds for the hired caregiver would dry up before you could move out, anyway. So learn while you can.
3) It'll be a somewhat better life for you, but not a fantastic one. Essentially what I am proposing is, you go out to work, you work, you come home, you're a part of the family and the dynamics and the caregiving scenario. A lot of work and not much less of a burden than you have now, but at least you will have the escape of work and the ability to answer back on any accusations that you're somehow not pulling your weight.
4) Making friends will also give you a support system and give you other perspectives. And, it will help if/when things really go South.
5) Working will also peel off some of the money going to you so that your father can use it for caregiving, so hopefully that arrangement would be able to last longer. Contribute to that if you can but don't feel obligated. It will be a help for you to be as financially self-sufficient as possible. Pay him back if you feel the need. But don't feel obligated to pay the caregiver's salary.
And, finally, look into social services. There may be ways to get your mother into some sort of program for caregiving that would be a bit less expensive, or perhaps get your father's debts at least partly forgiven. Call the hospital and doctors. They want some
money. They don't want to be stuck with a bad debt and no one paying anything on it, so ask! Say, "this bill is for $70,000. We are having a rough time of it and can only pay $10,000 (say this if you can pay $20,000 -- you are negotiating here, so don't open high). Will you help us by putting us on some sort of plan?" And see what they say. They may want some proof, receipts, pay stubs, whatever. But they may very well forgive a part of the debt. They want something on the dollar. You talking to them and working with them means they get more than zero on the dollar, which in this economy should have them jumping for joy.
Best of luck to you and yours.