I live in Georgia my daughters father left when I was 5 months pregnant. When she was born I filed for child support. Paternity was established and for the most part of the last 3 years (since her birth) has not paid or not paid on time, has some arrears, has gone to jail and court. Recently OCSS has garnished his wages and because of this he has been making regular payments. However he has had no contact for a year. I want to move out of state to better enrich her life putting her closer to family(I have none where I am). What are my rights on leaving the state, and can I file for child abandonment or do I need to close my case w OCSS to have full custody. I don't know what to do.(her father has a drinking and drug problem and I fear for her if he was to ever get visitation rights> HELP!

You have what ever level of custody that the family court gave you when you filed for custody. That does not change any him seeing or not seeing the child. And no this is not abandonment.

I would assume he has visits rights, if you wish to lower those rights, or ask for no vistis with cause, you do that in court.

In most states the custody papers state that you have to have the courts permission in order to leave the state. If your papers say this then you need to petition the court for permission to move. It doesn't sound like you will have any trouble.
Good Luck