Hi Patrick, I dont know the answer to your problem, but just to claify, the password is never stored in the app. When you enter your username and password in the app, it is sent to the Moodle server, which generates a Web Service Token and sends that back to the app. The app stores this token and sends it to Moodle with every request to tell it which user you are.
It's possible that this token is being deleted or expired at the server end, so when the app sends the token it asks you to log in again. You could take a look at /admin/settings.php?section=webservicetokens to see if it has an entry for your user.