Am i responsible for my spouse's credit card debt after they die in the state of Ohio?

Only in certain instances. If your name on the credit card debt? For instance, is it a joint account? If so, you are liable for it. If not, you are not legally liable for it. Some sneaky collection agencies will try to tell you that you are liable when you are not. This is illegal. Learn your rights against collection agencies by reading up on the FDCPA.