Unfortunately, I do not know the answer to your questions concerning creditors sending bills allegedly owed by your father, after he has died. Your father lived in New Jersey. Therefore,
New Jersey law governs the issues you raise: Is there constructive notice of death in this case, and are creditors required to consolidate their claims. Each state has its own individual laws and rules regarding creditors' claims following a death. I practice law in California. I am not at all familiar with New Jersey's laws and rules on these issues. Unless you want to do your own legal research on this (possible, but not easy), you'd have to hire a New Jersey lawyer to learn the answers to your questions.