Just came back from Korea last week. You would want a focus length from 18-200, so your kit and 70-300 would be ideal. Do take care in cold and heavy snow fall areas, if you are traveling in a tour group, the temperature in the bus is around 26-28 degrees and if you are planning to spend more than 30 minutes out in cold areas (0 to neg 0 degree) with snow fall, make sure you carry a clothes with you and wipe dry all camera contact areas of snow or moisture, make sure they are bone dry before entering the bus.
I had a bad experience of some moisture seeping into my camera (D300 & 18-200 VR), I suspect it is due to the difference rate of expansion and contraction of the camera body and rubber O-ring, thus allowing the moisture to enter the camera. The area I was in Jeju (Crater Mountain) was negative 14 degree with heavy snowfall and strong wind. It would not be a problem in extremely cold areas without snowfall.
Some images from my trip. Have fun and keep warm!