What parts of South Korea have the clearest sky without light pollution to see stars? It's clear the big cities have lots of light pollution, but even the places I have been told are "country side" still are relatively big cities. Are there any truly dark places?
Fortunately there's a website for this!
Lightpollutionmap.info visually shows you the most light polluted areas. Naturally it tends to be around cities.
The area along roads east of Chech'on and Andong look to be your best bets, that I can see, but a bit of panning around the areas you're visiting, you may have a chance. A car is likely going to be required to get to some of these, though. The Odae mountains between Chuncheon and Gangneung also look promising.
I would recommend the Juwangsan National Park, where you can visit the Jusanji Pond. This is where Kim Ki-duk shot his famous movie Spring, Summer, Fall, Winter... and Spring, hence on top of finding dark skies to watch the stars, you can enjoy the breath-taking atmosphere of the pond during day time.