You need to drag the songs onto the iTunes Library icon, NOT the playlists below that icon. Otherwise, the songs are added to playlists & iTunes will look for them from your LW download folder to find the song files to play. If you add them to the Library icon, you will see a small window showing you that itunes is adding the song & will be added to iTunes music folder.

Check out the sample gif images: How to transfer music to my iPod (& iTunes) {click on link}