Does a mutual agreement at the courthouse make a true custodial parent?

Before my wife and I moved from South Carolina to Texas about 4 years ago. She filled for child custody and support for her 8 year old (4 at the time). Her and her son’s father did not appear before a judge at the time. They met at the courthouse and had a mutual agreement, making her the custodial parent and awarded her child support which she only agreed to half anyway. Now the non-custodial parent (the dad) wants to take us to court and take custody. Is the doent from the courthouse good enough to hold him back? Can we file the case in Texas since there was never a judge involved in the first place or do we have to go back to South Carolina?