Are you taking your time to get to Florida and enjoying the sites along the way or you planning to get there quickly? If you plan to stop along the way I would recommend staying in Charleston, SC for a few days. James Island County Park and Oaks Plantation Campground are close to Charleston and are both nice. You can do Savannah, GA in a day, maybe two if you really take your time. Charleston is a much nicer city, but Savannah is nice too. We really like Skidaway Island State Park. Very nice park and only about 20 minutes from downtown Savannah. If you are into history, you can check out Fort Pulaski and Old Fort Jackson.
Further south in GA is Jekyll Island and St. Simons Island. Jekyll Island has a campground on the island. It's an okay campground, but the island is beautiful if you are a nature lover. You can also stay at Blythe Island Regional Park and Campground or Coastal GA RV Resort. Both are at the same exit of I-95 that Jekyll Island is and both are nicer campgrounds. Coastal GA might be hard to get into in January. They just started expanding the Resort, probably be complete this time next year.
We really like Hannah Park in Jacksonville. It's nice walk or short ride to the beach and a short ride to Jacksonville Beach, if you are into that thing. Downtown Jacksonville has some half decent Museums. We really like Jacksonville Zoo and there is also Fort Caroline.
St Augustine isn't too much further south. We like North Beach Campground right on A1A across from the beach. There are two nice restaurants in walking distance of the campground. It might be tough to get a reservation as winter is their peak season. Anastasia State Park is nice too, but again, peak season at Florida State Parks can be tough to get a site on short notice.
Coastal SC and GA will be cold in January. Day time temps are usually in the upper 40s to mid 50s. However there have been days in the 80s and days in the 30s. Being from OH, I'm sure any temp will be a relief :)
-Michael