They were correct that you do not have to pay assuming that you did not co-sign or guarantee the agreement with the credit card company (only being an authorized user of his cards doesn't count -- you would actually have to be a co-signer, co-borrower, guarantor or co-owner of the card account).

If that's true, they can only reach assets of his probate estate to satisfy the debt. Your home is likely not part of his estate because it likely passed to you by operation of law upon his death, i.e. it was titled in both your names as joint tenants with survivorship rights. Your joint bank accounts and similar assets would be the same. Any life insurance or investment/retirement accounts would also pass to you outside probate assuming you were the designated beneficiary.

so, unless there is money or assets in his probate estate, you don't need to pay the debt. Any amount forgiven could conceivable be considered income to his estate (but not to you) and you would simply need to file a tax return for his estate and his estate would be liable for any income tax. If you do get a 1099, speak to an accountant and/or attorney regarding all of the above and they can help file the appropriate returns and any other necessary administrative work.

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