Most of those springs are self-contained (Blue Grotto, Devils Den and Paradise Springs are the most popular) or are roped off or at least away from navigable waterways. The few that do connect to rivers or other navigable waterways (such as Ginnie, Troy and Rainbow River off the top of my head) you are required (but many people dont do it) to put up a flag.
Many other springs (Manatee, Peacock, Alexander and Blue Spring to name a few) do connect eventually to a river, but the state/federal parks rope off those access areas and you are not supposed to go past the rope (although it is all too easy when you are UW).
The key point is whether the area is navigable, if so take a flag, if not you probably dont need one. Now doing an OW course in one of the springs i often see a flag up - which the class use to ascend/descent with visual reference and kneel around the anchor of that line/flag, but this is something else altogether!
Great river drifts are at Rainbow River, the stretch from the devils part of Ginnie to where the tubes get out - if you have the air and the water is clearish, both of which require a flag. There is a short drift at Blue Spring (Orange City) when the manatees are gone, from the head to the swimming platform further down the run - no flag required.
I dont know about the panhandle, only north central florida.