The way iTunes and iPods are designed is not compatible with what you want to do. The best approach would be to open iTunes on your old desktop, and make a backup of your iTunes library, from which you can copy across all the songs from your old library onto your new laptop - not just the ones that were synced to the shuffle. One place to start for instructions on how to do this: http://support.apple.com/kb/ht1364
The other approach is to search online for a utility that will copy songs off an iPod without syncing it, add those songs to your new library, and then erase-and-sync the Shuffle. This way, you only get to save whatever was synced to the Shuffle at the last sync.