How about the Ocmulgee? It's a great canoeing stream from Jackson Lake down, at least the first 15 miles or so- that's all I've been on- but it's dam free for the rest of the way. There's class 3 rapids in the first stretch below the lake to the next bridge, and a class five in the second stretch that'll get you if you're not careful. Haven't floated it past there, but the fishing is great, and the river goes through or past lots of National forest and WMA's. The upper stretches flow pretty well, but I'd guess there would be plenty of paddling lower down.
I lived in GA when my kids were small, and we had many happy days on the stretch below the lake, fishing, swimming, and picnicking on all-day floats, with just enough mild rapids to make it exciting.