I got single dose packets of cough suppressant to take with each dose of the pills I took 3x a day, that seemed to work pretty well but I had no cough at all so far since surgery so maybe it was just that I didn't need to cough. You can always bring some with you just keep it under the airline max per container.
I find that the trick to clear the throat without using the vocal cords is to breathe out through your mouth like you are trying to fog a mirror. Try to make the sound haa not heh and drop your jaw while you do it. That is how I feel the least friction in my throat, you may be different. Now while doin gthat, just concentrate the breath into a short burst from the diaphragm like you got hit in the stomach but were ready for it. Do that over and over until you feel the urge to clear your throat go away, swallowing / spitting out anything that comes up.
I got the flu shot and while at the Dr. for that, I noticed they had surgical masks in the lobby to use if you had a cold so I grabbed a couple of those to use if I needed to. (I didn't)
My flight came in before 5pm Seoul time, so the free ride from the airport was available. If your flight is also before 5, you might be able to get that if it is still being offered.
Good luck Anjaq, don't be nervous