My sister, who has head injuries, encourages my mother to hate me. How do I deal with this?

You should probably step back now. If you've already tried talking to your sister about it and she still does it, then call APS and let them take it from there. In the meantime, let your sister and mother figure it out. If you still want contact with your mom then have it. If she starts accusing you on the phone and going on, end the call.

Melissa, I think that you are at the point of stepping back and letting the authorities intervene.

At 90 years old, living alone and easily manipulated means that she is vulnerable. APS will go check on her and get the proper agencies involved to get her the help she needs.

Let her make the accusations, they will check everything out and clear your name of wrongdoing. This is far better than living with the constant threats. Bring it on and be done with it.

Please do not block your sister. There are issues with isolation and cutting contact that scream of abuse, even when it is not there, these actions make it look likely.

Let mom and sister figure out how to get the groceries. While you are waiting for APS to respond to your call of a vulnerable senior.

Because you speak of your mother's phone as being under your control I assumed your Mother lived with you, however, your profile says that she does not, that you merely shop for her, and that you have no POA. Be certain that you NEVER accept POA over her, as this will only become worse. As she really has no income to steal, this is a bit ludicrous. Why not just hang up on both her and Sister when they get going. As to doing her shopping? No. Refuse to do so. Sister can do the shopping or Mom can order in. I would not have anything to do with even TOUCHING a debit card of hers, as it will only play into Sister's and Mom's own paranoia. Leave them on their own. Either take her shopping or let others do it.