Mid-July will be hot. DC gets very busy that week as many want to be in the Capitol for the 4th. If you're heading into Cherry Hill, arrive from the north in the early afternoon. Another option is Pohick Bay Regional Park south of DC. The sites have more privacy and you're more in the woods. Plus, for those days you don't want to go into DC, you can head south to visit the variety of Civil War battlefields around Fredericksburg. It's also close to Mt. Vernon. For a little long trip, it's two hours down to Monticello (really interesting). Only 10 sites with FHU, so reserve early.
You asked for other options. How about the White and Green Mountains in VT and NH? Cooler with lots of places to play in rivers and lakes to keep cooler. And as far as your concern about you rig at "higher" elevations, there are many camping options outside the core areas of these mountains that don't involve steep grades. Camp there and then visit everywhere else w/o the trailer.
Wherever you plan on going, it will be busy that week. For many places the 4th of July week is the busiest camping week of the year. Get reservations early.