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You need support from people who understand how debilitating this is. I know that here there are support groups for schizophrenia. I would call the local mental health center or talk to your medical doctor about what is going on. I would also get a medical exam. Stress can tire you out, and this letter sounds like you are under great stress.
You sound tapped out, LW. While everyone was worrying about your mother, no one worried about you. I hope you take care of yourself a little. You deserve the papmpering. Even the strongest can break.
I have no recipe for dealing with delusional siblings. Your sister has to get there for herself. She needs to respect that you need space right now. At some point she will get to where you and your father are. It's always hard to wait.
I wish you strength.