A Weapons Carry License is good for 5 years no matter where you might move to. It costs between $75 and $84 for new licenses, $30 to $34 for renewal licenses. (As noted you are not eligible to renew). A Weapons Carry License covers open and concealed carry of handguns and knifes with a blade of 5 inches or longer and made for offense and defense. The Weapons Carry License also exempts a firearm buyer from a NICS background check.
Applications for the WCL are submitted at and issued by the County Probate Court in your county of residence. Documents required are Photo ID and Proof of Residency (a Georgia Drivers License usually covers both requirements, but it may not so check before you go). Call your County Probate Court to find out what they will accept.
The application is provided to you at the probate court and you must fill out while there.