American Airlines has round-trip flights from Austin and San Antonio (AUS, SAT) to Seoul (ICN) for $877 with a connection. Flights February to April. Departures on Monday, Tuesday, Wednesday, Thursday, and Sunday. No Fridays or Saturdays. Extended weekends available.
A Carry-on bag and also a personal item is allowed for free on American. You can avoid paying for seat selection by accepting the randomly assigned seat at the gate.