For Gyeongju, you could fly to Busan and take a short KTX ride (20 minutes). Also, in the Busan area are the islands Geoje and Tongyeong, but that's a longer bus ride, maybe 1-2 hours from Busan.

That being said, the Busan airport is very inconveniently located. It's almost better to just pay the 90 USD and book a round trip KTX ticket.

If you have to transfer at ICN to PUS flight, the layover is around two hours. Then the flight is an hour. Then from the airport to Busan Station is around 30-45 minutes (30+ mins more if you want to go to Haeundae by public transit). KTX Direct from Seoul is 2-2.5 hours to Busan Station. Much faster and simpler.

Also, for your ticket, you can only do ICN-PUS as a transit passenger, so you'd have to go right when you land. Otherwise, you MUST do GMP-BUS...not sure what your award ticket will have for the stopover rules, airport changes, etc. on that part (to leave later).

Hope this helps!