He was arrested for domestic violence. He put his hands on me while I was pregnant with our 2nd son and he is dealing with that on going case. He broke the no contact and protective orders many times. He has a total of 5 warrants against him. One for domestic violence and the others for breaking no contact orders and harassing me. I have text messages from him stating he was on drugs such as cocaine.

I got a full restraining order in addition to the no contact orders. In the restraining order I was granted temporary custody of our 2 minor children and he was granted supervised visits, which he has not even attempted to arrange/set up.

I cannot afford this State of CT and would like to move where my family is in GA. Can I?

You have to apply to the court that issued the visitation order. If you can prove that he has made no attempts to visit the children, then you stand a good chance of getting permission to move. But you NEED the court's permission.

Yes you can