If I already have 5 old computers that I don't have access to authorized, I can't authorize my new computer in order to deauthorize the old ones. What can I do?

Use the Deauthorize All option and then authorize any computers you still have and want to use. See the following from Authorize or deauthorize your computer for iTunes Store purchases - Apple Support

If you are using a Windows PC to deauthorize all:

To deauthorize a computer that you don't have access to, you have to deauthorize all of your computers. Then authorize each computer that you have again. You can deauthorize all computers only once a year.

1. Open iTunes.
2. Choose Account > View My Account.
3. Sign in with your Apple ID and password. Forgot your Apple ID?
4. At the top of your Account Information page, go to the Apple ID Summary section.
5. Next to Computer Authorizations, click Deauthorize All.
6. Authorize each computer that you use again.

or if you are using a newer Mac, see:

To deauthorize a computer that you don't have access to, you have to deauthorize all of your computers. Then authorize each computer that you have again. You can deauthorize all computers only once a year.

1. Open the Apple Music app or Apple TV app.
2. Choose Account > Account Settings.
3. Sign in with your Apple ID and password. Forgot your Apple ID?
4. At the top of your Account Information page, go to the Apple ID Summary section.
5. Next to Computer Authorizations, click Deauthorize All.
6. Authorize each computer that you use again.