iTunes keeps wanting me to sign in

Hello trudy999 and welcome to Apple Support Communities. It appears you are having some difficulty with repeated requests that you sign in to the iTunes Store since updating. Try this:

System Preferences > Apple ID > Media & Purchases > Account > Manage

Sign out, reboot the Mac and then sign back in in the same place.

Cheers.

Suggestion - sign into the Music App with the ID Associated with your iCloud account and password and be done with it. If it is not allowing this sign in - that would be another matter.

If the Application will not launch at all - try restarting in Safe Mode - Intel CPU - SHIFT immediately at restart. Safe Mode does behind the scenes maintain things and loads slowly - Normal.

Run this mode for 5 - 7 minutes and test the Music App if it launches or not.

To exit Safe Mode - just restart in normal method.

Please re-verify as iTunes does not exist as iTunes since Catalina when iTunes was split into Music App. Podcast App and TV App.