It's important to understand this vote within the context of the political environment of 1856-1860.
Votes in these counties should be avilable from the Georgia State Archives, but you may not be able to find someone who can access them quickly. Georgia newspapers would have reported them, and of course offical vote totals should be recorded in the Secretary of State's election records filed with that office.
Some people writing about this topic seem to believe that political opinion is static. Based on Alabama newspapers printed during the summer of 1859, a significant number Southerners would describe themselves as Unionists. The same is almost certainly true of Georgia at that time. The U.S. Constitution protected the property rights of slaveholders, a bulwark against Northern opponents of slavery, so the Union served those who favored security, stability and profits.
The Harpers Ferry raid of October 1859 shocked the South, much as 9-11 affected the American public in our time. Following murders of slaveholders in Kansas in 1856, Harpers Ferry demonstrated that some Abolitionists would incite rebellion and bloodshed to achieve the overthrow of slavery. Northern sentiment in favor of John Brown, public mourning of his death, the Senate investigation into Brown's prominent supporters, all further undermined the Unionist position in the South.
http://xroads.virginia.edu/~hyper/contexts/kansas/jbrown.html
Even so, during the election campaign of 1860, the term "secessionist" was used to brand poltical opponents as unstable extremists. No one wanted that label, certainly not the Southern Rights Democrats. Secession was a last resort to be used if all else failed. As an aside, it's surprising to find that Southern newspapers have so little to say about Abraham Lincoln and the Republican Party. Depending on the kind of newspaper your decide to read, attacks are reserved for Douglas, Breckinridge or Lane and supporters of those candidates.
Along those lines, it's interesting to note that voters in Alabama's Winston County AL, remembered today as a Unionist stronghold, delivered a strong majority to the Breckinridge-Lane Southern Rights ticket.
As you mentioned, a county breakdown helps us understand the "local climate". For your purposes, it might be useful to compare county votes in the general election of November 1860 with those for delegates to the secession convention taken during December. The Unionist or conservative position may be more evident in plantation districts than might be expected.
Unionism as a factor in the state election campaigns of 1857 should be a useful benchmark for change during this period.