iTunes syncing (whether older version or current version of iTunes) is generally in one direction only, from iTunes library to iPod. You cannot use iTunes to transfer song files (and other content) from iPod to computer, except for items purchased from the iTunes Store.
If your old computer is still working (with the older version of iTunes), the best way to transfer your iTunes library is from computer to computer, like you probably did with the rest of your user data. iPods are media players; they not designed to be your iTunes data backup or data transfer method. If you transfer your complete iTunes library, you get everything (including supporting data such as playlists, ratings, date added, etc.). The same iTunes library will be on the new computer, not a new iTunes library with your old content.
This Apple document describes how to "consolidate" and back up your iTunes folder (with your iTunes data) to an external drive. You can then "restore" that backed up iTunes folder to your new computer, replacing its existing iTunes folder (with no content). When you run iTunes (on the new computer), it uses your iTunes folder from the your old computer, and it looks the same.
Manage and back up your iTunes media library - Apple Support