I think your issue is you're adding the songs to WMP's playlist whilst they're still in your LW downld folder. That's fine except that playlists are simply a list of the titles which point the player to the location of the song. If you later move the file, the playlist won't be able to find it. So the question is do you add the songs to WMP before or after you move them. If you're going to move them, then move them to a folder like a Music folder & then add them to WMP. See
Using windows media player for manually adding songs to WMP. WMP is simply a reference player (playlist player.)