Am I responsible for my deceased husband’s debt if I become executor of his estate?

First of all, an executor does not personally assume the debts of the deceased. The estate is liable for payment to the extent that is has assets to cover them. If the estate does not have enough assets to make repayment then the debts will either be extinguished as a matter of law (i.e. the creditors can no longer collect on the debts) or the spouse, in some states, may become liable for them (but not because they are the executor).

In this regard to this last point, one spouse is generally not obligated to pay the bills of the other spouse. However, there are several exceptions to this rule. If the couple lived in a community property state, a spouse would typically bear responsibility for such a debt (KY is not such a state). The second exception would be if the spouse signed or in some other way agreed to be legally bound for re-payments on the debt (you did not indicate that to be the case). The last exception falls under something called the "doctrine of necessities".  Under the doctrine, one spouse is liable for the "necessary" expenses incurred by the other spouse during marriage, such as medical bills but not car loans or credit card bills (anyway my research suggests that KY does not follow this doctrine).

Bottom line, it would appear based on the facts that you have presented that you bear no responsibility for your late husband's debts.