Divorce is granted, child visitation and support is set and being paid. But I'm unable to get the kids went the court has ordered, (1, 3, and 5th weekends week on week off in the summer)I'm in contact with them when she lets me talk to them. Can she file child abandonment against me? Also, I will be moving to another state in the next year or so, but will continue to pay child support.

I forgot to add that I live in Fayetteville Georgia.

No she can't. As long as you are paying court ordered support, there is no abandonment.

If she is not making your children available at the scheduled times take her to court contempt of court. Keep a journal documenting the times she violates the order.