fragment hunt quest said you need search three places, but when i talked with the king, he told waterspirit in a place, and the quest is completed, then i protected waterspirit but enemy took her and quest "the last battle" add.  so i don't search other two place which the quest said but story advanced. I guess may miss something, should i search the other two place first?

The other two locations aren't necessary to advance the story. If you go there, you'll find that the fragments are  already gone. The dialog with the King is the only way to advance the quest.
You can still visit those locations, if you want to do the riddles, but it's not necessary.