Thank you for the question. Only purchased content from iTunes will transfer from your iPod to you new computer. I see in your post that you have your old computer available. Home Sharing is a feature I recommend to migrate your media from one computer to the next. Below, I have included a couple articles that cover how to use Home Sharing and how to transfer purchases from your iPod.
Thanks for the reply. Most software for MP3 players have the ability to synch as a one way from the player into a library as a first synch. I cannot imagine what would happen if the original desktop or laptop was not available. Having to "re record" all the content on mine would take a few days! Why this is not available I am not sure, anyway thanks again for your interest and I will review the link.
Apart from purchases, iTunes will only transfer music in the one direction, computer to device. This is largely an obstacle designed to stop iTunes/iPod being used to illicitly share files. If your library is properly backed up as it should be there isn't a problem, or use third party tools to extract the media.