You will need to file a petition to establish paternity, and in it ask for an order establishing parenting time to place final decision making/custodial parenting in your hands.
My guess is that the father has filed the appropriate petition and is wondering what will happen if the Judge has to make the decision about legal decision making and parenting time. The judge could order the parents to go to an ERC, Early Resolution Conference where the parents could come to agreements with the help of a court officer. That would then be in writing and become the order. But the judge will want to make sure the child is safe and will want to know why the mother does not have her other children. "Full physical custody" would not really provide the mother with "parenting time" and that is unlikely unless the child is not safe with her. If she is unfit, the judge could order supervised parenting time. The public policy in Arizona is for "fit" parents to share parenting time and legal decision making equally. You might want to consider consulting with an experienced attorney. Judge can only make informed decisions with the information provided so it is important that the Judge knows all the facts and circumstances of each parent. I wish you the best of luck with this complex issue.