Wrote one message and lost it so will try again.
If the life insurance is payable to the estate then the bills should be paid from that as it is part of the deceased's estate. If the life insurance is payable to an individual than that individual collects the insurance and the bills do not have to be paid from it. If there is not enough money in the estate itself to pay all the bills, the creditors will have to write off their bills. As for the will stating all the bills should be paid, they are to be paid from the estate and they are not required to be paid by other individuals.
I would check with a lawyer to be sure that your state is like that as different states have different rules, but I think the above is pretty well general. The lawyer shouldn't charge much for answering a few questions for you I wouldn't think and then you will know what the laws are in your state for sure.
Just my two cents worth for the day.