When you transfer your account to a new computer - that doesn't transfer your purchased content. That just authorizes you to use your iTunes content on the computer. It is up to you to get the music back into your iTunes library.
If the songs are showing in iCloud, download them into iTunes and then sync them to the iPod.
You can also transfer the purchased songs into iTunes from the phone and then sync them to the iPod. This explains how to do it.