My personal favorites are the Dole Whip Float and the American Dream at the Fife and Drum in Epcot's America. Both of those would probably be classified more as ice cream though. The float is Dole Whip and pineapple juice, while the American Dream is vanilla ice cream mixed with strawberry and blue raspberry slush.
Honestly, you'll find options besides soda almost everywhere you go. All TS restaurants seem to have fancy non-alcoholic drinks of some sort. Just about all CS seems to tout some sort of specialty smoothie or iced drink of some kind, although they tend to change frequently. You'll find some sort of slush/icee/whatever you want to call them at almost every snack stand.