I inherited a car that has a balance due more than it's worth

Your name is not on the loan so therefore it would not hurt your credit. They could come after the estate for any loses though.

 

I am assuming that there is no will.

 

Your mother's estate owns the car and the associated debt. Not you.

Bring the car, the title and the will to the bank. Tell the bank (as the executor) that the estate can not pay the debt at all. Since the bank is not in the car business, they might ask what the estate can afford to pay for the car and retire the loan. If you do not want the car, do not make an offer. Offer the them the title and the keys.

You as the executor are entitled to fees for your services. These fees can easily equal the value of the estate, especially for smaller estates. You are entitled to get paid before the bank.