Your post is a bit vague. So what is it you want to do? (1) You want to transfer your music downloads to your new computer?
(2) Or you want to transfer your LW incomplete files to your new computer?

1. By default, your LW downloads are saved here C:\Documents and Settings\"your account name"\Shared
So that's where to look to find your older downloads.

2. Install LW on the new computer. Close LW. Replace the new incomplete folder with that of the old incomplete folder. It "must" include all files from within the older incomplete folder!!! This replacement must only be done whilst LW is closed otherwise it won't work properly.

Playlists work by showing the player where the songs are. But if the songs are in a different location, the playlist won't work. So locate all songs to where they were (identical location) on the new computer & your playlists should hopefully work ok.