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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:
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:
I hope this information helps ....
Have a great day!