For a parent to relocate to another state with a child, most states require either written permission of the other parent or permission of the court. As it appears that you have primary physical custody, it should be easier for you to obtain the court's permission than if you have joint physical custody. If you want to move without the child, you can do that; however, you will lose custody and will likely have to pay child support to the father and pay for the travel costs for your visitation.
I would advise seeking advice from an experienced family law attorney in Georgia before moving.
Mr. James' answer may be correct in his state, but is incorrect as to Georgia.
Whether you can move and whether you will have violated a court order depends on the terms of the court order (and parenting plan) in your case. Note that if you do move, he can force you back into Georgia's courts if he files a case within 6 months of your move.
It is vital that you see a lawyer to discuss this before planning a move.
No, you will not automatically lose custody of your kids. However...
Most order require that a parent provide notice to the other parent if there is a change in their address and/or phone number. Some (very few) require permission prior to moving. The permission provision is rare, but violating it is contemptible. Therefore, you should consult your custody order to verify that you're not required to obtain permission. If you are required to obtain permission prior to moving, you will need to file an action with the court to request it. If you're not, just provide notice.
Despite moving, you will still be required to make the children available to the father at the times provided in the order. Of course, you living in NY will likely make the terms of your order difficult to abide by. Therefore, the father will have the right to request a change in custody/visitation due to your move. Whether primary physical custody will transfer to him will still depend on the best interest of the children. However, if you keep primary custody you will also likely become responsible for most of the cost of visitation (since the increase in visitation costs will be your "fault").
The above answer is a general explanation of legal rights and procedures. It does not constitute legal advice. Nor does it establish an attorney-client relationship between the individual posting the question and the attorney providing the answer.
Sign up to receive a 5-part series of useful information and advice about child custody law.