American Airlines has round-trip flights from Dallas (DFW) to Seoul (ICN) for $872-$888, NONSTOP. Flights February to March. Extended weekends available.
The lowest price $872 will require booking two separate one-ways; probably not worth the hassle. AA website will price this at $888, Priceline will price it at $881.
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.