I live in North Georgia and am needing to find out about child abandonment laws concerning my ex-husband.

In GA abandonment laws allow for his arrest for non payment of court ordered child support. There is not statewide same response, some counties in GA will act on this better than others.
So if he is being on support, you contact the DA office and the GA child support enforcement department.

As for as custody and rights, GA does not allow taking away a parents rights for non visit and non payment of support.