Do you owe a personal debt to an estate if there was no written agreement?

Yes, if the debt was a legitimate debt as many, but not all, unwritten debts are. The only difference when it comes to an estate is that the standard of proof in court might be higher when suing an estate than suing the decedent when he/she was still alive. The reason for this is that the decedent is not available to give his/her side of the story and raise any denials or defenses that might be appropriate. Most states have laws that require such a higher standard of proof, so these have to be checked to see if this rule applies.