Moving the kids out of state

The basic facts are this: He left in April. He signed a paper agreeing to pay child support and a year of alimony. He quit his job in October to be with the new girlfriend and her baby. He is now unemployed, staying at his parents with the girlfriend and baby having extended visits, not spending any time with our 5 year old daughter. Maybe those are more than basics. I started working in June at a minimum wage job and do not make enough money to pay the house, electric, water, child care, phone, household supplies and gas in the car. I have a child from a previous relationship and his family - father, stepmother, grandparents - are very supportive of me and my kids; however they live in TX. My husband and I moved from TX to NC and it is all his family here. My mother-in-law is my child care.
The question: Can I legally move from NC to TX and take my children. Neither one of us has any money, but I know he will fight me. What does the law say? Also, if he has my daughter in his custody and refuses to return to to me what rights do I have to her? Man this is all so frustrating. Sorry - thanks and I look forward to your prompt response.

You will be at risk of your ex filing an emergency order to have his child returned to the jurisdiction of NC. Unless and until there is a custody agreement or court order in place both of you have equal rights to the child. Your best course of action is to file an action for custody and fight to have the court order that you are the primary custodian, and that it is in the best interest of the child to move to Texas with you.