Well, there are several other FATE games out there besides "The Dresden Files". There is "Spirit of the Century" for a bit of twenties and thirties pulp and serial-movie adventure, "Legends of Anglerre" for fantasy, "Bulldogs", "Diaspora" and "Starblazers Adventures" for sci-fi "Agents of S.W.I.N.G." for spy-fi. It doesn't take much to adapt "Spirit of the Century" for adventures on Barsoom, twentieth-century weird sci-fi and horror, or Girl-Genius steampunk. I suppose that it is probably partly because FATE is so simple and easy to adapt that we are seeing so many differ settings of FATE show up. So if you like Aspects and Fate Points you aren't at all stuck in the Dresdenverse.
I'm going to echo what Patrick Crusiau said above. Getting rid of detailed or complicated rules won't by itself make freeform roleplaying appear out of nowhere, and for that matter you can very easily conduct story-telling jam sessions with a detailed and number-heavy game system as your safety net for emergencies. I've done that for decades.
I suggest that what you want is, as you hinted, to come out of the dungeon-crawling fantasy genre. There are less opportunities for role-playing where the conflict is open and violent, and players are naturally more inclined to try storyline stuff and take a chance on character over pragmatism where the danger is less immediate and where killing all your opponents is not so easy to get away with.
Players tend to be drawn to things they know about and methods that have reliable support from the rules. I suggest that you look for a game that has rules for sneaking and guarding, infiltrating and patrolling, evasion, pursuit, deceit and detecting lies, dissimulation, acting, seduction, persuading, intimidating etc. etc. that are at about the same level of complexity and reliability as combat. The easiest way to draw PCs into any sort of activity is to give the players a feeling that it is about as reliable as combat. That could mean vague and simple rules for everything, but it could also mean detailed and complicated rules for everything. When the chips are down, players turn to where they know the rules.
Another thing that I suggest is to pick a genre of adventure fiction, TV, or film that your players are familiar with and that gives them a clear idea of the wide range of possible actions, of character types, and of stock situations. I like Westerns, "hard-boiled" detective stories, and the "James Bond" style of highly-romanticised spy story as "weaning campaigns" to easy dungeon-crawlers into roleplaying. But if your players are "Star Trek" fans, run "Star Trek".
