I've no way of logging out of the app.

Tried uninstalling and then reinstalling and all the bookings came back.
Had a look through my phone and can't find anything to do with AA unless it's some obscure file or cookie hidden somewhere.

Added:
On my Samsung Galaxy S8+

Make sure the app is installed.
Go to settings, apps, click on the app, storage, clear cache.

That seems to have done it.