Davenginny, Hope you enjoy your visit to Ohio! Before we settled down in a seasonal site, we camped in the state parks and really enjoyed them. Our favorites were Indian Lake near Russells Point, Salt Fork near Cambridge, Deer Creek near Mount Sterling, and Dillon near Zanesville. All are within an hour to an hour and a half from Columbus. They have nice sites with plenty of room. Indian Lake, Salt Fork and Deer Creek have really nice lakes if you enjoy boating, water sports, or fishing. Dillon also has a lake but I wouldn't get in it or eat anything caught from it.
I agree with the previous poster about Mohican - very nice park. One word of caution with Alum Creek, it's one of the busiest state parks and the sites are smaller so you hear more noise. We enjoyed Indian Lake so much we found a seasonal site and we're there every weekend. I'd be happy to answer any questions if you're interested in these parks. Again, hope you enjoy your visit!