My mother is in much debt--if she passes away and her estate doesn't cover all bills, can credits come after her adult children?

No.  Unless you have agreed to take over a debt, you have no contractual obligation because your mother owed a company money.  The creditors do have to be paid, before any money goes to the heirs -- and if it isn't so, that's when the creditors can go after the children, for the money they got from the estate.

But if you get nothing out of the estate, you're not required to put anything in, if there isn't enough for all the bills.  The creditors are just out of luck.