How do you move to a new town? It's always greyed out at the train station.
Also is there any way to turn on player weight gain on an existing save?
The 'Dizzy's Delightful Doughnuts' upgrade lets you raise your current town's obesity rate. Once it hits 99%, you'll unlock the ability to move to the next town.
Right now, there's no in-game way of enabling player weight gain on an existing save. But you could export your save data (via Extras on the main menu), then edit it and re-import it to change this.