First of all do not log out -- no need to. I have app on an iPhone , iPad and Kindle.working fine. I just signed ouf the app on iPad. The only way i saw to sign out was to say it was not me signed in. After signing out I signed back in. All I was asked for was my email address and Amazon password, I did so and was back in the app. and had no issues issues.
Suggest you remove the app from your iPad and reinstall it.