Same, exact problems since the V 3.4.6 update.
I followed your suggestion and went into the Notes section on lds.org account. I deleted the trash (had done so several times before over the last couple of weeks) and deleted any folders I had created under "Notebooks" and "Types." The only things remaining are, under "Notebooks," "All Items" and "Unassigned Items" which I assume are defaults. Under "Types," the only thing remaining are "Highlights" which, I assume, is a default. All my annotations are highlights (I have some notes associated with those highlights but all notes are first marked with a highlight). "Trash" indicates "(0)."
ONE BIG ITEM: I performed a test. I logged out of my lds.org account and then deleted the Gospel Library app from my iPhone 6 Plus. I then reloaded the GL app to the iPhone and signed into my wife's lds.org account. It worked flawlessly for the few minutes I experimented with it. It immediately synced and in the GL app's "Settings" window under her lds.org username, it indicated when the last sync occurred. I tested the sync by highlighting a scripture on the iPhone and verified the change appeared on my wife's lds.org account. I then highlighted a scripture on lds.org on my Mac and the change synced over to the GL app on my iPhone. So, it appears the app is working fine with my wife's lds.org account. I then logged out of my wife's lds.org account and deleted the GL app from the iPhone. I next reloaded the GL app onto my iPhone and logged into my lds.org account. The behavior is still as before; when the app tries to complete the initial sync with my lds.org account, it seems to stall. The GL app's Settings window is still indicating "Syncing Annotations" and has for 10 minutes now and, if the app's behavior is the same as before, it will continue to indicate that for hours to come. This is the same odd behavior as before.
It would be easy to say that there is something wrong with my lds.org account but it is equally plausible to think that it is the syncing interaction between the GL app and my lds.org account that is at fault, i.e. the app's fault, I mean.
Despite the fact that it may still be a problem with the GL app, who do I contact to find out about someone checking my lds.org account for an anomaly? Is that something that can be done?