Rick & Brenda,
If you're looking for fhu campgrounds, Cross Creek (as Mike mentioned) or
Berkshire Lake Campground are both a bit over half an hour north from Upper Arlington. Both are located off exit 131 (head east for one, head west for the other) at I-71. That exit has both Pilot & Flying J travel centers, several fast food restaurants as well as a Cracker Barrel (an rv-r's fav!).
If you don't need fhu's, both Alum Creek State Park and Delaware State Park will be good places to stay. Delaware SP is another 12 miles or so north of Alum Creek SP. Although there are fhu's at Alum Creek, there's only 3 of them & they all have less than 20 ft pads. As Mike stated, Alum Creek SP is just down the road from Cross Creek, so you would have the access of the travel plazas & stuff at that exit. The rest of the sites at these state parks are electric only. If you want recommendations for sites at either of these state parks, let me know & I can get you a short list to choose from.
Most of the other campgrounds in the central OH area would be a farther drive for you or ones that I have not been at to recommend.
L-