Can't authorize MacbookAir

"Can't authorize MacbookAir: Apple won't let me authorize this MacBookAir because it says I have more than five devices. When I look at my listed devices under my Apple ID, it just shows five, including this MacBook Air.[...]"

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5 Computer Limit:

You can only have 5 computer synchronized with your Apple ID. So, you'd have do sign out of iCloud on another computer.

Deauthorize All Option:

Try Deauthorize All. You can only Deauthorize All items once every 365 days. So, you may have to wait up to a year before you can perform this, if you have no access to any of these computers.  Go Here: Authorize or Deauthorize your Computer for iTunes Store purchases - Apple Support

You said that you couldn't find Manage Devices in the Music app, perhaps it should be Account Settings? You can see the authorized devices after clicking on it.

I also found an official guide which may be helpful to you. Here is the link: Authorize or deauthorize your computer for Apple ID purchases - Apple Support Maybe you can try reauthorizing your Mac to see if the problem can be solved.

"Thank you. What worked was an online chat with Apple support. They were able to deauthorize all, and I could then authorize the current devices."

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You are most welcome.