Usually you can back up music from iPod to a new computer/iTunes with "Transfer purchases". That requires your apple id.
In your situation, you can try some third party tools for music transference. I recommend iTransfer. It's a Windows based tool for file transference, a tool that helps you transfer music files from iPad to computer hard disk or iTunes without logging in (need assistance from iTunes).
The iTransfer tool is not free but you can get its free trial version.