feedback Nolichucky is under 750 cfs, doable but not especially challenging.
French Broad still is running about 1500 cfs, which is starting to get on the low side for a raft, but still reasonable. Much under 1200 cfs, I would find something else. The Pigeon is reliable, short enough and easy enough shuttle to do multiple runs. Release schedule during the summer is Tuesday, Wednesday, Thursday and Saturday. Since this is one of those odd runs where they are generating when releasing, so they often run on unscheduled days if they have water. Oh, scheduled releases don't start until 11am.

The Ocoee is a bit of a drive, but an option too.