if adult children can no longer deal with their parents, can a third party be designated as their POA to take over financial decisions?

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If you need to take a break perhaps the authority you have as your mothers POA allows you to hire a bookkeeper or CPA or some professional to handle her finances at your direction and supervision? At least for a year or so to see if it will help. Think of ways to lighten the load and not have to interact directly.

But, with dementia they do. It is part of the disease. We all need to remind ourselves of that constantly and not take anything that is said personally. Has her doctor been made aware of her behavior. There are meds that would probably help, though as I have found out just today, that is not always the case.

Unfortunately no parent should ever treat their child like dirt.

When it comes to Power of Attorneys, only your parents can make any changes or appoint someone new.   Would either of them understand a legal document if presented to them?   If not, sorry, nothing can be changed.  If there are other family members, maybe someone would want to try to be their Guardian, but that requires a Court progress and can be expensive.

Sounds like your Mom might have dementia.   Please don't blame her for what she is saying because sadly her brain is not working correctly.   The "home" she wants to go back to is probably her childhood home where she was happy as a child.