what happens if I cannot deauthorize iTunes on a Mac I am recycling?

Hmm.

Start in the iTunes menu bar, click Account > View My Account...

When prompted, "Sign in to view account information" with your Apple ID and password.

You should be presented with an Account Information window. If a different window is presented, again click Account in the menu bar and select View My Account...

Hopefully you will now see the Account Information window with the "Deauthorize All" button at the bottom of the first section.

This is from my High Sierra laptop with the iTunes app, but the process is exactly the same in Catalina with the Music app.

You might have to pull the right edge of the window to the right to enlarge it and make the button visible.

Anyway, if this still doesn't work on that Mac, you do have plenty of additional authorizations to use as you noted, so maybe you can do this at another time from your new Mac.

Then at the very least you'll have one less device available for sharing.

Hello, bernadette52.

It's no problem. From iTunes or Music app you can click the Account menu and login to your account with your Apple ID and password. There you'll find your Account Information and under the heading Apple ID Summary should be the option to Deauthorize All. Use that to deauthorize all of your computer activations.

Don't worry, the next time you use a Mac to access purchased media you will be prompted to authorize that Mac and you can do so again then. It just takes a moment. 👍🏽