Dallas is AA territory, and AA doesn't charge ridiculously high amounts of miles to fly international business class on non-AA metal (at least not yet), as UA does. So I at least don't think collecting AA miles is a mistake.

Chase Ultimate Rewards points transfer directly to Korean Air's ff program as well as UA. One option might be to book two one-way tickets, one using AA miles and one using Chase points transferred over to either Korean Air or UA. Right now I'd just concentrate on making sure you get the sign-up bonus for the Sapphire Preferred card; you can figure out what to do with the points later.