As mentioned, you'll need to reapply for Georgia Medicaid. Many doctors close to a state line will accept the plans from the neighboring state but to be sure you'll need to contact your doctor.
Medicaid is a state program. Each state has its own program. If you cross the state line, you will lose the Medicaid card that you now have, and you will be in a state of limbo while you fill out application papers to apply for Medicaid in a new state. Many Disabled people have ended up in a big mess by crossing a state line and ending up ill in a hospital and finding out that they had zero medicaid insurance. It is a bad situation.
No. Medicaid is on a state by state basis. Once you move out ot TN, you no longer qualify for TN medicaid.