How do I recover purchased albums from an old Apple ID on a new Mac?

Apple went and changed my Apple ID when they changed everything to an iCloud account. Then they deleted iTunes. I used a Windows machine when I purchased the albums that has long since gave up the ghost. I switched to Mac computers 5 years ago. I came across the albums on some old jump drives that I had downloaded them on and transferred them to my Mac music library but they say the computer is not authorized to play them. I tried re-authorizing but that does not work either.

How can I recover these albums?

[Re-Titled by Moderator]