You likely should review your facts and options with an attorney located where the probate was.
Generally, the law requires valid and properly filed claims against an estate to be paid before distribution. The person appointed by the court to handle the estate generally has a duty to follow the law. Failure to follow the law may result in personal liability for that appointed person.
With a house, the estate likely had enough funds to pay off the $10,000. Whether the bank had a valid claim against the estate may need to be looked at. Whether the bank has a personal claim against the appointed person likely is also an issue.
You should review your facts with an attorney. There are many issues here that no one can give you an opinion since there are many missing relevant facts.
Ordinarily the credit card debt would have been paid off out of the estate before any distribution was made. In your case it sounds like you may have assumed responsibility for your mother's debts. If you did you have to keep paying. Consult an estates and trusts attorney before continuing payments.
Once again Mr. Kaman proves that he should stick to criminal law. If you were in California, the debt should have been paid out of the probate estate. Since it was not and since more than one year has elapsed since your mother's death, the debt would be uncollectible in California. Check Michigan's law (if that's where your mother lived) and see what the statute of limitations is on debts belonging to a decedent. Alternatively, if an attorney assisted you with the probate, call him/her and ask whether you need to pay it.
You need to speak to a Macomb County probate attorney. Your answer depends on whether the credit card debt was filed as a claim against the state and whether you notified this known creditor during the probate proceedings. There are two issues: 1) is the estate legally liable, 2) if so, is there a practical solution with the credit card company where they may accept less than the entire amount in order to resolve the matter. I'd be happy to help you find a local attorney who could assist.