I've done several trips on the Suwannee. I made some pretty long posts a few years ago, but not sure if I can find them again. The upper Suwannee, from the Okefenokee swamp down to south of White Springs is probably best. Further downriver from there, it gets bigger and more crowded with boats, people etc. We never had problems with finding spots to camp along the way.
Suwannee Canoe Outpost in Live Oak has good info. We used our own canoes, but had them take us upriver and drop us off. They have several different spots they take you to based on how long of a trip you want. They have a pretty good chart on their website about the water levels. The river can be entirely different based on the level. Too low and it's slow going with some areas actually too low to paddle. Too high and the current can be so fast it can be dangerous.
Another really nice trip we did was the Blackwater River out west in the Panhandle. It's the onle complete sand bottom river in Florida. It averages only 2-3 feet deep and there are sandbars perfect for camping at almost every corner.
Blackwater Canoe Rental will take you up river and drop you off at a spot based on how long you want to stay. I only did that river once, but have wanted to go back ever since.
Edit: I found a couple of the old posts...
https://www.ar15.com/forums/hometown/Another-canoeing-camping-trip-on-the-Suwannee/10-552865/