Here's a message I sent to a person with a similar plan, but on short notice so it doesn't have many options and excludes a lot of what I hinted at above (but still some great roads..)
You'd take GA400 to get started, thru Dahlonega to the Rock Pile (Intersection 60 & 9/19).
Tight and twisty:
60 to Suches (Int. 60 & 180/Wolf Pen Gap)
then 180/Wolf Pen Gap to US129
OR
Fast and twisty:
9/19 to US129 at Turners Corner
US129 over Blood Mountain
Let your GPS take you to Hiawassee, GA, then Franklin, NC.
If at all possible, stop at Cafe Rel in Franklin for lunch. Unless you have the tastebuds of a goat, you won't regret it. CASH ONLY!
From Franklin, set GPS for Deals Gap, which will take you via Hwy 28.
It's about 3hrs from I-285 to Franklin, and just over an hour to Deals Gap from there. You could get three runs in (a "sighting run" up, then fun runs back down and up again) and then head towards Knoxville (use GPS).
Note, although the Dragon is technically 11 miles, the last 2 are not very interesting, so 99% of people "running" the Dragon stop and turn around at "the overlook" which is just over 9 miles from Deals Gap Resort. You'll see a sign for "scenic overlook" (or similar) and there are pull-offs on both sides of the road.
Would suggest you take Foothills Parkway when you get down the hill - you'll run alongside a huge lake for a while, the turn is 6 miles from the overlook, just past a bridge over a creek (there are signs). It adds 5 minutes to the trip and as well as being another great road, there are some great views as well. Overlook to Knoxville is just over an hour.