Bob -
Great that you are doing the trip! Here are a few of the places I have stopped on previous trips that have turnaround room. I am a memner of the Holiday Inn Express frequent stay program (called "Priority Club"), so they are all HI Express places I'm listing below.
Florence, SC, which is about 30 miles into SC, has a couple HI Expresses, but only one has the good turnaround. It is the HI Express at I-95 Exit #170. If you have a GPS, punch in 2101 Florence-Harllee Blvd., Florence, SC. Monster parking lot with lots of room along the right side for RVs, etc.
About another 51 miles down the road there is a HI Express in Manning, SC. It's a little tight, but the parking lot does completely encircle the building. You can usually find a row of unused spaces on the outer edge. This is an older place compared to most HI Expresses, and it has exterior door entrances, but it is not bad, and it was pretty cheap to stay. It's off I-95 Exit #119 in Manning, SC. The address for a GPS is 2201 Raccoon Rd., Manning, SC.
About another 111 miles down the road is Hardeeville, SC, near the bottom of SC. If you make it to Hardeeville on the first day, you're in really good shape. It's only about 120 miles to the FL line from there. There is a Holiday Inn Express at I-95 Exit 8 in Hardeeville, SC. The parking lot does NOT encirle the building, so don't try to go around it. But, there is lots of room in front. The hotel is set back behind a gas stattion / convenience store, and you can circle around the back of the convenience store, which circles past the front of the hotel. The best place to park is to the right side of the parking lot (right side perspective as you are facing the hotel). The address of this HI Express is 145 Independence Blvd., Hardeeville, SC.
Regarding driving through the Washington, DC area, I go out to Warrenton, VA on Rt 29. and then take US 17 to Fredericksburg. That makes sense for me, as I am already starting quite a bit west of Washington. It probably wouldn't make sense for you. If I were you, I would consider taking US 301 down the Eastern Shore, then cross the Chesapeake Bay Bridge, and take 301 down thorugh MD until you cross the 301 Bridge over the Potomac to Dahlgren, VA. Follow 301 south to I-95, and you will get onto I-95 well south of Fredericksburg and all the traffic mess of Washington, DC area. Look at that route on the map, and you'll see what I mean. Under any circumstances, do not try to go through the Washington, DC area during rush hours.
Have a great trip. Let me know if you have more questions. I'll be out of town tonight, and might or might not get back on line. Either way, I'll check in by later on Sunday.