I saw you added a SWH while I was typing.
Okay, if the baby's father is an abusive drunk and you want to keep yourself and the baby away from him, then just submit the form without his name. You will not be able to collect child support*, but he won't be able to come after the baby without having a court-approved test either.
*If you and the baby are on any kind of social services, the state will usually require paternity to be established.
You should probably seek legal counseling too. If he does try to get visitation because of his mother, you may need to go to court to protect your child by request supervised visits.
Original answer:
You don't need him to sign the form to turn it in; you can leave Father blank for now and do a correction later. You do need to try to formally establish paternity though.
A man that doesn't believe a child is his and wants a test will not willingly support the child, and the state won't pursue him until a paternity test is done.
In your shoes, I'd have had the test done before leaving the hospital.If there is no one else that can be the father, I don't understand why you haven't done it in all these months. I'd be ticked off and anxious to be able to shove it in his jerk face.
If you've been delaying because there is a chance he isn't the father, it's time to put on your big girl pants and face the music.