Hi Eunice, welcome to the forum. My condolences to you and yours for your loss.
As far as I know, the only way that you are responsible for the debts on the credit card is:
1.) If you were a joint card holder with him, implying both of you are responsible for the debt(which as I understand it, is not the case).
2.) If you live in a community property state. But I am speaking with no knowledge/experience or education in community property laws.
On the surface it appears that collection agency/credit card company, is in violation of the FDCPA(Fair Debt Collection Practices Act), by dunning you.
I'm sure others with more experience than I will chime in and contribute. We have many very experienced and knowledgeable people on this forum.
Good luck to you.