I have been split up with my daughters father for years, and she lives with me. He seems to be looking for a job in another state. I have never taken him to court for anything. We try to work things out on our own. Is there a way to have legal papers drawn up without going to court to establish custody, if both parents mutually agree?

Legal papers require a court approval. I suppose yoyu could draw up an informal agreement and have it witnessed & notarized. That might be considered binding if you have to go to court.