iTunes syncing is generally in one direction only, from iTunes library to iPod. You cannot use iTunes to transfer song files from iPod to computer (except for songs purchased from iTunes Store). If the iPod still had your songs, you could use a third-party method or utility to transfer from iPod to computer, but in your case, your iPod was erased when you synced a mostly empty library with it.
Therefore, you'll need to transfer your complete iTunes library from your old computer to your new computer, which is actually the best way. If your old PC is still accessible, with its iTunes data intact (or you have a backup of your iTunes data), please post back for more details about doing the transfer to your new computer.