For some reason, I did not see that you had replied to my reply (usually get an email notification), so sorry about the delay.
iTunes syncing is generally in one direction only, from the computer's iTunes library to the iPod (and that is by design). You cannot use iTunes to transfer song files from iPod to computer, except for songs purchased from the iTunes Store. However, there are third-party methods and utilities that can transfer from iPod to computer. If you do an Internet search on something like "ipod transfer," you should get some links. (The CNET article seems to be popular.)
Once the song files are on your computer's drive, you can add them to your new iTunes library. After confirming everything is there, you can then sync your iPod to that iTunes library.