If you are added as a joint obligor you have responsibility. That usually requires your signature to do though. If he added your name to the account as a card user you do not have any obligation for the debt.
His passing does not affect the above. You are not liable for the debts of another unless you promised the creditor that you would be liable. this is done by signing a promise or other agreement as a cosigner or surety.
The next time you get a bill write "deceased no estate" on it and mail it back if indeed your spouse did not die with any assets attachable to an estate in probate. if you are unsure about your spouses estate you need to consult a probate attorney to make sure you are not acting fraudulently.
I am sorry for you loss.
Generally speaking, you are not liable for your late husband's credit card debts. If you signed for the authority to accumulate credit card debts on those cards, the answer may be different.
If you were not a co-signer or authorized user, and if you never charged anything on the credit card, then the money owed should come out of your husband's estate.
God bless you.