Back in the early iterations of the traverse, I did most of the intent of it in my 2wd Ranger (and sections before in my 2wd Escape) - not Tray mountain, admittedly, and not all in one trip - probably four or five trips all told. I used the old pdf versions and mapped it with google earth, printed the maps. Yeah - old school, but I don't like using hands-on technology while driving (had rather have a big map in the way of my vision of the FS road, I guess).
I intentionally wandered off the track in several spots - spent my first night at Three Forks off Warwoman Road, for example, since it's one of my favorite places on earth, and drove up into the Chattahoochee WMA just north of Helen one afternoon to do some trout fishing. That's where I almost got stuck w/o 4wd.
I hammock camp, carry a small tent with the theory that sometimes I can't hang (but never have had to use it), so am traveling pretty lightly. I never had an issue finding a campsite, though I did get off the track to sleep at a favorite hangout from when I lived in Athens and wandered north GA pretty much every weekend.
Anyway, my main advice is to check the current (as of the day you're going) road closures by the state DNR or National Forest, as a LOT of the unpaved sections go (went?) through Wildlife Management Areas, so may be closed for a two or three day or week long hunt unless you are drawn for that hunt. Later in January is less likely to have that issue, as it will be post-deer season. Some of the public campgrounds are closed in the winter as well.
You are supposed to have a WMA permit (or Georgia Lands Pass) to be in a good many of the WMAs, which is not cheap for non-residents, but is good for a year, at least. The list of WMAs that require the pass (or a hunting/fishing license) is on the Georgia Wildlife site (not allowed to post links yet)- looks like the Chattahoochee National Forest lands are exempt, but if you go into some of the others for a free overnight spot, you might need one.
Wear your long johns unless it's a freakishly warm winter :)