You need to copy over the iTunes Library.itl file from the old computer to the new computer. This file has your playlists in it.
It typically goes in the Music\iTunes folder. You may already have one there if your music is intact.
After placing the file, try to play a track, if it asks you to locate the track this means your music is in a different location than it was on the original computer. Hopefully you left your music is the same organization structure when you moved it. Anyway, click on the locate button, then navigate to where that song is on your new computer. It will connect it. If your organization of music is the same, all tracks should link up.
Then you should be able to use your playlists, and sync without erasing your iPod.