I really like Lake Nona; it's more interesting than most Fazio courses seeing as Mike Strantz was involved in its construction.
Southern Dunes is my favorite public play in greater Orlando.
If you can get Streamsong to work, you won't be disappointed.
More towards Stuart, I am a big fan of PGA Village, in Port St. Lucie. The Dye Course is highly underrated among his work. It's one of my favorites in the state and I try to play it at least a couple of times a year.
Closer to Stuart proper, the best non-private option is Florida Club, by a pretty good margin IMO. It has its own Dye influence; Dick Gray, a longtime superintendent who helped build Crooked Stick, laid out FC. It reads as solid, low-profile Dye pastiche.
There's also the reversible nine-holer at the Martin County course, Sailfish Sands, which is good fun.