But now, the Gear Advisor starts popping up, trying to get me to put the old 360 version Ring #1 into Ring #2's slot, since iLvl 360 is better than 350. Except I can't do that, because of the "Unique-equipped" modifier. So I click "Cancel". And then the window continues to pop up after every action my character takes, over and over and over, until I get rid of the old ring by vendoring, scrapping, or destroying it.
As late as possible. Early on, you’ll want to follow the missions in Battle for Azeroth right up until you get the option to send a follower on a two-hour long quest. As soon as you get there, stop doing the War Campaign missions. You can come back to them once you have reached something like 119 and a half and finish leveling with those if you want. If you don’t use these missions to cap off your leveling journey, then they’re the first thing you’ll want to do once you reach max level.