As a DPOA for my Mom can they come after me if bills don't get paid?

Only if your name is also on the account (utility bill, cable, whatever).

As a Durable Power of Attorney, you pay the Nursing Home first according to the amount agreed upon, which is usually her entire SS check. Other bills like mortgage and utilities do not get paid, nor are you responsible personally, as long as you wrote POA after your signature. DPOA's can get in trouble only if they misdirect the funds they are responsible for.
On the other hand, with credit cards, if you were a co-user of a credit card account, you will have to pay.