At the beginning of The Hobbit, Thorin has this sword, right?
Then he finds the goblin-cleaver. Where does his old sword go?
Well, I happened to notice this (that’s Kili, btw, badly screen-capped):
So, my headcanon (or actually canon?) is that after finding the goblin-cleaver, Thorin gave his old sword to Kili, who is mainly an archer and therefore didn’t have a sword with him.