Augusta sits on the border of GA and SC.
The following SC State Parks are close to Augusta and have RV camp site.
Hamilton Branch State park (just over ½ hr away)
Baker Creek State Park (across the lake from Elijah Clark, see below, 1 hr away)
Hickory Knob State Resort Park
Aiken State Park
Barnwell State Park
See http://www.southcarolinaparks.com for more details.
On the Georgia side of the Savannah River we have:
Mistletoe State Park (one of the closest to Augusta, cottages, 45 minutes )
Elijah Clark State Park (also have cottages)
Bobby Brown State Outdoor Recreation Area (over an hour drive)
See http://www.gastateparks.org/ for more details