I like both sweeter cocktails and more traditional, less sweet ones, so I'll do both when at Disney - depending on the mood and time of day.
Favorite bars for me are
Geo 82, Territory Lounge, Nomad Lounge, Geyser Point, Crockett's Tavern, Art Smith's Shine Bar, Frontera Cocina, Jock Lindsay's, Dahlia Lounge, Topolino's Lounge/bar, Victoria Falls, Abracadabar, River Roost, and Outer RIm.
For the non-tropical drinks - Geo 82, Territory, Geyser, Crockett's, Shine Bar, Dahlia, Victoria Falls, and River Roost. For the more tropical sweeter ones, the other bars in that list.
There are a few spots that have really good drinks that are on the sweeter or tropical side but without tasting like pure sugar bombs, or having a combo sweet/tangy taste like some margaritas can. Frontera has some great margaritas, the Leaping Lizard at Nomad is pineapple and cachaca (a less sweet Brazilian rum alternative), River Roost's blood orange margarita is lovely and more tangy sweet. Jock's Cool Headed Monkey is sweet but has pineapple and tangerine to give it a bit more tang too.
Geo 82 does the best Old Fashioned, though Territory's is quite good too. And always remember, most of these are bars that can prepare any standard cocktails you want - you don't have to order the ones listed on the menu.