If I have full sole custody of my son and his father and i were never married can i move out of state against the non-custodial parent's will?

YES YOU CAN . once you have full custody you have the right to move wherever you want to that's the reason you file for custody right...the the reason full custody was ordered to give you full permission to do whatever you want ..the court gave u full custody only because they see you are a fit parent at the time of filing. however the father can file for full custody in the future but i doubt he will get it since the child bee living with you for so many years and you were the sole provider