My mother, of almost 80 years, is sneaking around and behind our backs opened up a new bank account and then proceeded to give out her routing number. What should we do?

Has your mother been evaluated for dementia? I wouldn't tell her that is what a checkup with her doctor is for, but you probably should have her checked out by a specialist. You could alert the doctor by letter, and then take her in for a regular checkup, if she is stubbor about any issues that point to poor decision making.

If she checks out okay for cognitive issues, she has a right to do with her money as she pleases - even though it's alarming to you. The only way to stop this could be legal action. You may want to talk to an elder law attorney.
Carol

Seek guardianship immediately before it gets worse. I had the same problem with my father who is 81. Make sure you find a good elder attorney for this.