If I have the father of my unborn baby sign away his rights he thinks I would have the right to go after him for child support. Am I right that he wouldn't have to pay child support?

No, signing away rights only gives up every ounce of say in the child's life, you still have to pay child support...

To elaborate on what Al said, signing away rights only means that he will not have any decisions regarding the health and well being of your child. He will still be financially responsible to take care of the child.

That basically means that if there is a medical emergency (just as an example) and you want the procedure done, then he cannot protest it. That is just one example.

Why would you not want child support? Are you afraid that it would create some kind of closeness you do not want him to have? Or give him an opportunity to be in the life of you and your baby?

First nothing at all can be done till the child is born,

Next he is correct, except in cases where there is a new partner to adopt, signing away "rights" does not stop obligation of child support. Even if you did not ask for any, if you or the child would latter receive state welfare the state can go after him for that.

No, you're not.