One of my toons has been stuck showing a transmog look I haven’t had on her for about two weeks now. I’ve tried logging in and out on different toons, tried logging in and out on Battle.net, tried logging in and out on here, tried logging her out in a capital city, etc.
Nothing seems to work. 
Short answer, no.
Long answer, Nooooo.
You’ll just have to wait until the armory catches up.
It’s been weeks now though.
Don’t worry. I heard Blizzard is going to upgrade the Armory from using Dial Up to ADSL at some point in the future.
You make that joke, but I know there are big companies that still use DOS. Worked for some of them.
Oh lord, now I feel old…
I know, right?
I’ve found that loading up a real stinker of a transmog set will cause the armory to update immediately and hold onto that set for at least a week after you change out of it.
I use the Better Wardrobe addon and I’ve found that if you save your current look then that’s what character screen toon will look like.
You could try changing appearance at the barber, which is another recommendation I’ve heard before…but, honestly, the armory seems to just update randomly sometimes. 
Nope, didn’t work.
I’ve had my armory update twice in the last 24 hours while fiddling with transmog, not sure what causes it to happen when it does.
For a week this toon had shoulders that I had immediately transmogged on equipping, but the armory displayed them as not transmogged. I tried everything and was unable to update it.
It’s happening in game too. I’ll tmog something and log out to see the old gear pieces still showing. I think it’s the game. Something isn’t updating between the servers.
It could be connected to this memory leak…
Blizzard said it was affecting DX11, but I’m forcing the loader to use DX12. The loader wants to launch WoW in DX11.
This is not really a new thing. It has been happening at least since the start of Dragonflight, possibly even back during Shadowlands. It hasn’t been unusual at all for quite a while for me to look at a character, ask myself “didn’t I transmog that?”, log in to fix it, and find that the character screen was wrong.
Spending x amount of time in game on that character seems to do the trick. How much is x amount of time? 
No clue. Must be more than 2 hours.
I played one of my old toons for 3 hours last night and it still has the same mog I applied months ago. I don’t get it lol.
Well if you didnt change the tmog, it would still have the same look. 