When you first import content from Dropbox or another storage service provider, you're prompted to sign in. OnSong retains a token provided by the service for future communication and refreshes that token as needed. However, if you want to sign out of a service/content provider, the process is pretty simple.
Open the Songs Menu by tapping on the Songs button in the Menubar and then make sure you are viewing the Songs tab. Next, tap on the + button to open the Add Songs Menu where you can add songs to your library. Tap on the storage/content provider you would like to sign out from such as Dropbox. At the bottom of this menu, you'll find a Sign Out button or a door icon with an arrow. You can tap on this icon to sign out of the service and be taken back to the Add Songs Menu. The next time you enter the screen, you'll be prompted to sign in.
You can view a complete list of storage and content providers that are directly integrated into OnSong in the Add Songs Menu in the user manual.