I don't think you need any campaigns going, I never touched any BFA content and I can easily go to Kul'tiras with either Horde or Alliance.
I think the issue is that the same boat in StormWind looks exactly like the DragonFlight one which can mix people up. Just wait till the right boat comes, as the captain shouts about it.
Edit: Unless you're at the start of the DF campaign that revolves around the black scale thing you get from your inventory which can cause phasing issues, finish that up untill DragonFlight area, then return back and wait for the Kul'tiras Boat. One thing of note, boat porting is very buggy so you might crash countless times and be stuck on a loop.