Like many others we head south in January. If you are coming down the I81, I77, I75 corridor you should be out of freezing weather during the day by the time you get into the Carolinas, unless we are having unseasonably cold weather. If so you should have daytime temperatures above freezing in Georgia. As others have indicated if daytime temperatures are above freezing, and nights are below freezing, you can safely dewinterize and stay overnight in your Airstream. The furnace generates enough heat in the belly pan at night to keep your pipes from freezing.
Traveling down the I85, I75 corridor northeast of Atlanta at Winder, Fort Yargo State Park has a delightful campground and is open all year. Stone Mountain Park, on the east side of Atlanta near the I285 beltway is nice but rather expensive. South of Atlanta, High Falls State Park is very close to I75 and a comfortable place to overnight.
When we travel down the I-95 corridor we often overnight at the Walmart parking lot in Walterboro, South Carolina not far from the Georgia line. We have stayed there several times with no issues.
Quite a few Airstreamers heading to Florida in early January (January 8-10 in 2016) meet up at the Canopener rally at Topsail Hill Preserve State Park in the panhandle. This year over 100 Airstreams are signed up for the event and the park campground is currently completely booked. However there are always a few cancellations in the weeks before the event so watch the park reservation site and the Canopener thread on Airforums if you think you might want to attend. In January 2012 we attended our first Canopener as new Airstream owners and have been back every year since. It is a great event and all Airstreamers are welcome. Here's a link to the Airforums Canopener thread:
Airforums Canopener 2016 Thread
Warning. The weather can be chilly or delightful in the Florida panhandle in January. Last year we had below freezing weather at night and daytime temperatures in the low to mid 50's but everyone still had a great time.