It will depend on the airlines: whether they interline, ie handle each other's luggage. If they do, you will be able to check your luggage all the way to Fukuoka, regardless of whether you have one or two tickets. Been there, done that. If they don't, tough luck: you'll have to pick up your luggage, pass through Customs, and go upstairs to check in again.
The main issue here is of course Customs: Korea has an automated x-ray system for luggage offloaded from planes. Anything suspicious will lead to the luggage to be tagged with a huge yellow padlock that plays a loud music when arriving at the Customs checkpoint. A bunch of pills might (or not) trigger the alarm.
However, even if you get tagged, you should be fine. My luggage was tagged once, because of an excessive number of alcohol bottles, and upon showing my onward boarding pass for Japan to the Customs officer during the secondary inspection, I was let go without a fine, and my bottles were not confiscated.
Be polite, speak slowly and softly – their English is more often than not VERY basic (I speak fluent Korean, EXCEPT in such cases ;-) – and the officer will be very reasonable too.