Actually, just before departure is one of the best times to find award availability in all classes, unless it's a true high season (like Golden Week in Japan). I'd suggest if you are stuck buying a last minute refundable fare, you keep checking availability for award travel right up to departure. And if it's non-refundable, pick it on a carrier where you can get an upgrade award and ask right up to departure time.

I don't know who flies to Seoul that partners with Amex. Delta certainly doesn't, and SQ (which goes out of its way to not permit award travel) would require travelling via SIN which would add many, many hours to the trip.

I'd also suggest you contact a consolidator ASAP.