Oxfordpete, since nobody has given you the "very-specific" ID, time-dating, and manufacturer info on your button, here's the info.
Although post-1865 GA State Militia buttons exist, yours was definitely manufactured during the civil war era. The book "Record Of American Uniform And Historical Buttons" (by Alphaeus H. Albert) shows your button as #GA-4-A1. The "small details" in the emblem on yours match GA-4-A1 exactly... and I can see enough of your button's backmark to tell that it says "Horstmann & Allien NY" with a circle of rays around the loop. That backmark was manufactured only during 1860-65, according to the McGuinn & Bazelon on button-backmark dating.
My early years of civil war relic-digging (the mid-1970s) were at Atlanta Campaign sites. I can tell you that civil war Georgia State Seal buttons are very hard to find in Georgia, because most of the Georgia Militia regiments who wore them were serving out-of-state, with the Army Of Northern Virginia and the Army Of Tennessee. So, double-congratulations to you, first for finding that civil war GA State Seal button, and second for managing to dig one in Georgia -- which the vast majority of relic-hunters there have been unable to do.