Also having a R bike, I remembered that the swingarm bearings/bushings can get loose or worn. The R way to check that is to center stand the bike, have someone hold the rear brake down and then attempt to wiggle the back wheel (should work for a K as well). With the rear brake on, the only looseness that should be felt is that in the swing arm - ero, cause of the tire wear/strange handling. If there is no "wiggle" while holding the rear brake on, I would look at the final drive as my cause.
My 0.02 worth
;D