There is a way to use an iPod with multiple computers. See this page for more information:
http://docs.info.apple.com/article.html?artnum=61675
You could try clicking on Store and then Deauthorize computer until it says that this computer is not authorized. Once you get there, try authorizing again and see if you can get your songs on the iPod.
Apparently, logging into iTunes and purchasing a song does not mean your computer is authorized. I just deauthorized mine, tried to play a purchased item and received the message that my computer was not authorized. I logged in to iTunes and downloaded a song, successfully. I then attempted to play it, and, lo and behold, still getting the authorization message. I was, however, able to subsequently authorize the computer without any problems.
CG