Have tried relogging several times, as well as logging into other chars. Both my husband’s char and I are level 58, but the Dragon Isles Await quest is not popping up at all.
Have stood in Org, relogged in and out several times, exited WoW and Battle.net completely, took the zepp to Dragon Isles. Nothing seems to be helping it pop-up for either of us?
Same issue on level 59 character
Did you check your mail?
Nothing in mail.
I am also having this problem on my level 58
Same. I have watched the quest auto delete on login for all characters that I did not run through the pre patch questline.
It appears you need to be level 60 to get the quest to go to DF.
Or that is what has shown up on my GF leveling her character. At 58 she didn’t get the quest but when she was 60, it popped for her.
It would be great if this could be patched/ adjusted. A bit obnoxious to change a level requirement that’s been a given from the other expacs with no notice. 
Even now you can google tons of guides, from reputable sources, that also detail the typical 58 requirement, which just makes me feel like the change should have been telegraphed. (Or, it’s a bug to be fixed.)
Would love some confirmation is this is intended or not. From what I saw on the beta, 58 was the requirement. So I only leveled up my characters to 58.
We finished leveling up to 60 and as soon as we stepped into Org, we got the quest.
Same here. Would love to know if this was intended and then they decided to not actually inform folks, which is pretty (insert stronger word than crappy here.)
(Unsurprising, but still.)
A blue response would be fantastic. This is documented precisely nowhere, and sites like IGN are quoting the level 58 requirement when it does not seem to be the case.
Seconded! A blue clarification would be great. And not just an acknowledgment that it’s intended, but why it was changed.
This is a strange incongruency - I’m beginning to think Blizzard has no idea what they are doing
Only just beginning? 
Well, okay then.
You don’t receive a quest; all you have to do is to go to the Stormwind Harbor to where the Dragonflight ship leaves and get on it without the starting quests. that’s all; did that on my warrior at lvl 59.
Nope, just hopping the boat did not provide any quests to begin anything in Dragonflight. I could get there on the boat at 59, but no starting quests.
Absolutely must be a Level 60 for questline to open!
If anyone cares, you can still go to the Dragon Isles even without the quest. Just stand on the edge of the airship dock that sticks out from the ship, and it will take you there regardless of your level.