If I become executor of my deceased grandmother's estate, will I be held liable for any unpaid bills that she may have accumulated?

That would be a no.

As the executor of an estate you are not obligated to pay the obligations of the deceased from your own assets. You are obligated to pay the obligations of the estate and the deceased from the estate assets before any distribution to an heir or legatee. If you improperly distribute estate assets to an heir or legatee before paying a proper estate debt or creditor of the decedent you could be liable.

No. Her estate is liable. If her estate doesn't have enough funds to cover the liabilities, then there is a process for paying claims in part.? Make sure you have an attorney help you. If you do it wrong, then you can make yourself liable for mistakes.