Am I responsible for my dead husband's hospital and doctor bills?

No one is responsible for someone else's debt. You would only be responsible if you cosigned for any debt incurred. However, his estate may be responsible. Any assets he owned at the time of his death could potentially be used to pay bills. If your assets were intertwined with his, you may want to speak to an attorney to clarify what assets need to be probated and what may be subject to his medical bills.