If you had a card on your father's credit card account that you used before and after he passed away are you responsible for the debt in New York State?

This is a serious issue. In every state upon the death of an account holder the card issuer(s) must be given timely notification. Even before that is done the card is NOT to be used. Person's using the card(s) after the account holder's death can be held responsible for any/all debt incurred. They may also be liable for civil and/or criminal penalties.