Probably not within your 20 miles beyond the I-285 loop, but the Georgia International Horse Park in Conyers is a little known RV park.
From their web site:
We’ve been told by many of our visitors that the Georgia International Horse Park is one of the best kept secrets that they have found with regard to RV parks east of Atlanta. The GIHP offers a full-service RV park for the convenience of our guests. Fully equipped with water, electric and sewer service, these 140+ sites are located near the Stable Complex and the Walker Arena Complex. Full-service shower facilities are also located within the RV Park and their use is included in the daily RV site rental fee. Due to limited capacity, you must call in advance (770-860-4190) and reserve your space. GIHP office hours are Monday-Friday, 8:00 a.m. – 5 p.m. We look forward to your visit!
You do not have to be associated with an event in their park to use their campground. We stayed here for a couple of nights several years ago. Nothing fancy, but full hook ups and very quiet.
Here's there web site:
Georgia International Horse Park websiteYou can find the campground under the "other amenities" tab.
Fred