interesting.
My guess on the syncing component has maybe to do with updating happing or coming on the backend process or service where things sync across the various GL apps. I have a feeling that it is being worked on. I don't know any details about that, but just a guess. Hang in there just a bit longer on that component.
As for your Settings button, you may want to reset the app within the Windows Setting section (not the GL Settings area) /Apps and features.
Depending on the Update version/build you have of Windows 10, you may be able to reset the app.
That is basically done by doing the following.
Windows 10 app reset
Open the Settings app (looks like a gear) , and reset a desired app. Here's how to do that:
Open Settings.
Go to System > Apps & features.
Find an app you want to reset, and click on Advanced options.
Just click the Reset button.
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Once you have done that, open up the Gospel Library app and then should be prompt to sign in with the LDS account as the setup.
You may want to close the Gospel Library app before resetting it.
Hope that helps in at least the GL app setting component on both the production and beta app.