Depressed need advice

When I was 26 weeks pregnant my baby’s father left the state I’m in. I was going through a divorce that never got finalized an since that day my husband and I have worked things out. The baby’s bio dad has dropped contact with me an told me he wasn’t planning to take anything to court and he didn’t want to deal with it. He’s only bought one thing of diapers and wipes and drove them down to me (he lives 6hrs away) he hasn’t checked up on baby or myself for two months. My husbands signing birth certificate and is taking on the child as his own...what rights may real baby daddy have? I don’t think he will come back but I’m getting major anxiety as baby is due any day.....have any of you ever had the situation like this? How would custody work with him living 6 hrs away and dealing with a NB baby? It’s giving me a great deal of anxiety and making me nervous baby daddy is sooo unpredictable please help