You don't have the original computer anymore. But surely you still have the disk that contained the program you used on the old computer to manage the playlists? Then install that on your new computer and you should be ready to go.
But it could be you have to restore the data that that program needs (the PC copy of those playlists) from your backup. That's like iTunes and iPhones. If you synchronise your iPhone with an empty library on your new PC with iTunes, everything on the iPhone is deleted. The Sansa program might do something comparable. Why not check the user manual and the help?