Am I responsible for credit accounts in my spouse's name only afterdeath?

In the State of Ohio if you did not sign any loan documents, then the debts are not yours.  Starting there, however, is only a start.  The question is your husbands estate liable?  The answer could be yes.  However, if he had no estate, then you would not be responsible for his debt. Given the question, the answer is uncertainty until you seek a consultation with a bankruptcy/probate attorney.  It will take a consultation to create a plan of action because of status of the estate, if any, and the totality of your financial condition.