ADVDropbox.Logout works at the level of the component using the REST API to get access to DropBox files and this is unrelated to a login/logout scenario in a separate browser instance.
When you perform logout, it should forget about your access token and when you want to perform the AdvDropbox.Connect to get access to your files again, prompt again for authentication & authorization (and thus allow to login with a different acocunt to authorize a different user)