Unable to authorize iTunes purchases on my new MacBook Air

I purchased a lot of iTunes music over the years. Most of it is on an older laptop. When I try to transfer a tune to my newer laptop I get a message saying the computer is not authorized. I authorize the computer, but when I try to play the tune it tells me again that the computer is not authorized. I suspect this has something to do with an older Apple ID that I updated, but not sure. It's frustrating.

[Re-Titled by Moderator]