I live in Texas as well, the heat it awful. I've also been able to find really good nonprofit and religious organizations that offer help to people in your situation. You don't have to be apart of any congregation you just have to need help. If I were you, I'd take advantage of that. If you want to give your partner a chance that's great, but don't do it because you feel trapped.
He still has rights even if you don't list him. But he would probably have to go to a court to force you to do a DNA test on the baby to prove it first. The baby deserves to know his dad even if you don't like him.
Talk to the midwives about it, I dont know about where you live but they have programs here to help out women in your situation.