Hi JTassa,
Welcome to the Support Communities!
If you have questions about your iTunes Store account, you have access to your entire purchase history and can review any issues with the iTunes Store Team:
http://support.apple.com/kb/ht2727
The article that you referenced contains the detailed steps that you need to get your music onto your new computer. The best scenario I've found is to back up the music from your current computer to an external drive, and then add the music to the iTunes Library in your new computer. Then follow the steps to authorize your new computer for your Apple ID and sync your iPod to it.
I'll highlight some of the important links / notes from that article for you below:
How to use your iPod to move your music to a new computer
http://support.apple.com/kb/HT1329?viewlocale=en_US&locale=en_US
Download the latest version of iTunes. You'll need to authorize any iTunes Store content. If you won't be using your old computer, you can deauthorize it.
iTunes: Back up your iTunes library by copying to an external hard drive
http://support.apple.com/kb/HT1751
http://support.apple.com/kb/HT1473
I hope this information helps ....
Have a great day!
- Judy