I'm liking Waterdeep: Dungeon of the Mad Mage Preview...

Me and my players have been playing the preview of this module as if it was a one shot, and so far we've had two sessions where we were still having a lot of fun while traveling into the first level of the Dungeon. I'm thinking of purchasing the adventure module after we've finished the preview, but I had this one question: If I decide to buy it, what happens to the game that has the preview mode? Do I have to create a whole new game and have everyone find a way to transfer their characters into the new game?

Your existing game will continue to exist as it does now.  You'll need to create a new game with the DotMM as the base Module, then migrate the players into that game.  You can do that via the Character Vault.  If you have other things you want to move over (custom maps, handouts, etc), the only way to do that is with the Transmogrifier, a Pro Subscriber Perk.