Although you stated that you had not synchronised the iPod with the new iTunes Library when this problem first occurred, is sounds to me as though you possibly did. But if you haven't set Sync options between the iPod and the new Library, that might explain why only the music from the store is on the iPod, after having downloaded directly from the store.
So, check the Sync options between your computer and the iPod.
ThomasJNewton wrote:
So I synced the iPod, took hours. Eventually it finished and everything was on there. Then I turned off the Mac and went to play something. Immediately the song I was trying to play disappeared.
This too sounds as though you were playing music on your iPod that was stored on your computer, not on the iPod (don't know how though). Perhaps you were looking at the content of the iPod from your computer rather than directly on the iPod, I don't know.
Check your Sync options because if you are now using a different Library (on a different computer) that's what you need to do.