Dispersed Camping above Upper Chattahoochee
I admit, I'm not a fan of most established campgrounds - the "official" campground is $16/night. It has potable water. Opens April -October. Sites vary from tent-walk-in, to some spots that are spacious and more private to others tiny, out in open with little shade in summer. Some sites are along the creek. I walked thru the closed campground to access Horse Trough Falls in March - a lot of trees had come down, so the FS had been chopping up the trees - many sites had lots of wood piled up!
However, if you are a bit more rustic, there are a number of dispersed sites before you get to the Upper Hooch. Bring your own tables, water - but there are large (free!) spaces to camp.
Regardless, be bear aware. I saw 2 cubs and fresh scat on my last visit. They ran from me - but no need in tormenting them.