I try to open iTunes, but get a dialog saying iTunes is updating. I'm stuck

You might try again in safe mode, or see this section from iTunes for Mac installers - Apple Community:

If you want to install an even older build then you can try this approach to remove iTunes:

1. Open Finder > Applications > Utilities > Terminal
2. In Terminal type cd /Applications/ and press enter to open the Applications directory
3. Type sudo rm -rf iTunes.app/ and press enter to delete iTunes
4. You will be prompted to provide your administrator password which you should enter to complete the process

You may need to temporarily disable system integrity protection (SIP) while you are removing iTunes.

tt2

Well!

The Mac Installers page recommended going back to version 12.6.5 (or something like that) of iTunes, which would install over later versions. Then I could install the last version compatible with High Sierra, which is the OS running on my 11 year old iMac.

Installing 12.6.5 (or whatever) gave me several dire warnings, then failed. However, that effort seems to have put the fear of God into iTunes, which is now running fine.

Thanks for guiding me through this!!!

Force restart the computer. Try again.

tt2