How do I file an extension for my Georgia partnership?

No separate extension is required for GA as long as you comply with GA instructions:

"A FEDERAL EXTENSION WILL BE ACCEPTED AS A GEORGIA EXTENSION IF: 

(1) THE RETURN IS RECEIVED WITHIN THE TIME AS EXTENDED BY THE INTERNAL REVENUE SERVICE, AND  

(2) A COPY OF THE FEDERAL EXTENSION(S) IS ATTACHED TO THE RETURN WHEN FILED. "

See the GA instructions and extension application at: https://dor.georgia.gov/sites/dor.georgia.gov/files/related_files/document/TSD/Form/TSD_APPLICATION_...