TL;DR: The fix is to go into party sync, pick up the legion starting quest, finish it (skip the intro), complete the quest in Dalaran, and then leave party sync.
I had the following problems on a fresh character (leveled in BfA):
- Couldn’t use the Dalaran Hearthstone, it would instantly cancel the cast
- Thrall would tell me he has an urgent message, but not give a quest
- I could do the broken shore intro, then start the Argus entry, but once in Orgrimmar, I couldn’t progress as the lady on the boat would stay grayed out
- the entry quest (The Hand of Fate)
- (To Be Prepared) quest wouldn’t phase me to the legion version, so I couldn’t complete the 4 requirements for the legion intro to be flagged eligible for Legion stuff
After completing the legion entry via party sync, everything started working normally again.