I just heard back from our lawyer, he said:

To answer your question, an unwed mother does have sole legal and physical custody. The unwed father has no rights until a court grants them to him. He can sign an ROP, but all that does is give him the right to petition for visits or custody. Rights don’t spring biology. So you can—and should--deny a father the right to take a child in your care, unless you have written consent from custodial parent. It’s best to have written statements from unwed mothers stating who they authorize to pick up their child. If a father has no custody rights, the mother can still grant visits at her own discretion.

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