It is not unusual for spouses to have the right to use the credit cards of the other, and usually this is done with the permission of the creditor.
Death does not automatically forgive the right to collect debt. In most states, the probate process is used to pay off any claims for debt that are filed on time from any assets that belong to the person who died. If the spouse or someone else were a co-signer on the debt, the creditor could look to the co-signer for payment.
Hope this perspective helps!
I would add to the prior answer that if there are no assets from the estate then there is no fund from whcih to pay the debts nor any remedy for the creditor.
I want to note that life insurance proceeds are not subject to the bills of the decedent unless the insurance is paid directly to the estate.
It is not fraud for a spouse to use a credit card of the other spouse for their own purchases, absent any other facts indicating fraudulent intent. The spouse that actually made the purchases is certainly responsible for them, but the credit card companies only occasionally pay attention to that.
While I have trouble imagining a "moral" obligation to pay a credit card, the user should feel free to pay for what he bought. However, I am absolutely disgusted by creditors who call up heirs of insolvent estates and allege that there is a "moral obligation" to repay a decedent's debts. It's simply not true.
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Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.