Would it be possible for me to just buy my ticket using my Korean name and passport but when I enter the US just explain my situation and show my US passport to show that I’m a US citizen?
Yes. If you have any documents that connect your Korean name to your US name, it might be a good idea to bring them along, but I doubt you'll actually need them.
Your major problem, I suspect, will be getting the airline to board you without ESTA authorization on your passport. If you can convince them that the two passports both belong to you, however, you should not have a problem.
You may also want to try applying for ESTA with your Korean passport. We have had a few reports on this site of ESTA approval for dual citizens even when they disclosed their US citizenship on the application. If you have ESTA authorization for your Korean passport, you won't even need to show the US passport to the airline.
When you arrive at the US immigration checkpoint, show your US passport. If any questions arise about the flight's manifest, you can explain that you booked the ticket in the Korean name, and even show your Korean passport if need be.