Sounds like AA at its "best"

Hopefully you saved a copy somewhere of the app message somehow which should be more than enough proof that AA messed this one up big time.
Hopefully you'll get relief in Korea since this one sure looks open and shut to me (non lawyer)
Also don't forget to file a complaint with the US DOT since AA did manage to break a couple of laws and regulations over here as well.

Hope everything has worked out so far and good luck.