Tallulah should run this spring (as stated above, first two weekends of April). It was shut down in the fall due to a dam gate that was damaged in a high water event in August, but they were supposed to have had that fixed by now.
The Green should be running those first two weeks of April, and the Cheoah also runs on those weekends.
The Ocoee flume is broken, so it could also be running a good bit then.
We've been getting a lot of snow so far this winter, so as long as it stays reasonably wet, we should have a strong enough water table that it won't take too much rain to get other things going while you're here.
As we say in the intro to the Southeast chapter in our new River Gypsies' Guide to North America
, the first two weeks of April are probably the best time to travel to the Southeast.