Asked previously
Answered previously, although I am an attorney, I am not licensed to practice law in Georgia.
That said, any unmarried male (alleged to have sired a child) has no parental rights in the child unless and until the male has gone to court and established paternity; more importantly also requested and a judge ISSUED a court order describing custody and visitation rights.
As of this moment, according to your recitation above, the male in question has only affixed his name to a birth certificate.
There is NO existing custody order, because he has failed to go to court to seek one.
Therefore, YOU and YOU alone, the mother of the child possess sole and exclusive custody over YOUR child.
If that is true, you are free to quickly and silently remove yourself and your child to any state or territory of the United States of America.
If you so desired, as of this moment, you and baby could scurry away to any of 125 odd civilized nations across six of seven continents on planet Earth.