Yes, the taxes do get paid first. Once the money runs out, a "final" 1041 is filed and you're basically done.
I hate to make certain creditors wait another year since I really don't want to be nagged by collectors.
Once the money runs out, when you receive a notice from a creditor, you simply send them back a statement with a copy of the death certificate. Remember, "you" are not responsible for the debts of the deceased. But you are responsible for the management of the estate. Once the final 1041 is filed, the estate is dissolved.
The only way another would be responsible for any debt of the deceased, is if they co-signed on the debt. But that's not always true either.
