My condolences on the death of your husband. The answer to your question is, probably.
When someone dies, their credit card debt becomes a debt of the decedent's estate. Probably the agreement between your husband and the credit card company said something like, interest will accrue on unpaid balances. If that's what it says, then interest accrues until it's paid off. The credit card company files a proof of claim with the estate, and the administrator pays the claims of the estate out of the estate's assets until the assets are exhausted.
That's not legal advice as I don't hold Kansas licensure. If you need legal advice, contact a Kansas lawyer who handles probate matters.
Max is right. It will greatly depend upon the agreement your husband had with the card issuers. But, three things to keep in mind:
First, if you are not an authorized user of the account, then you are not obligated to pay the debt personally. It is a debt that must be paid by your husband's estate, but only if the card companies properly and timely file their claims.
Second, anything your husband purchased with the cards (as well as anything else he owned at his death) may be used to pay these debts (if the claims are properly filed).
Third, sometimes there are death benefits associated with credit cards that could pay the outstanding debt and/or give the estate cash after the death of the card holder, but you must make ask they issuing company about death benefits. They often do not volunteer this information.
All Kansas decedent's estates are required to be handled by an attorney.