DC in winter is cold. Bloody cold. The bonus is that visitor numbers are low and you can get to see some of the iconic sights without the crowds. Good idea to head south. So long as it isn't actively snowing, driving should be fine. Just remember that ice forms on bridges first, so if it's close to freezing, be wary.
Richmond, as the Confederate capital is full of history. The Civil War stuff was fine, but what really sent chills up my spine was to stand in the church where Patrick Henry gave his famous "Liberty or Death" speech.
Heartily concur re Charleston. Amazing place, though it certainly won't be at its best in January. New Orleans, likewise, but it should be warm. There's a place on Mobile Bay beside the Interstate east of Mobile. Felix's Fish Shack. Right on the beach, awesome seafood. Bourbon Street in New Orleans is great for people watching, but not great for the kiddies into the evening.
Memphis is fabulous for some great Southern sights. Graceland and the ducks at the Peabody. Beale Street can jump.
Disneyworld in Florida. And Kennedy. Both places that deserve a day. At least.
Dallas for shopping, Fort Worth for tourists. The Stockyards has a longhorn parade twice a day, with cheesey cowboys and sherrifs and baddies. The Kimbell is one of my favorite art galleries and the Asian Gardens just down the road a haven of tranquility.
I love road-tripping in the USA. Check the internet for dining recommendations, so you aren't stuck with the chains. Oh yeah, and you have GOT to eat at an Outback Steakhouse at least once. The Blooming Onion is about a bazillion calories, but soooo tasty!
The Interstates are good for quick and boring driving, but the "blue roads" will get you closer to the real small-town America.
Have fun and post lots of happy snaps.