Cannot sign in to account on Music App

I've recently begun having a problem being unable to sign in to purchase music from the iTunes store in the Music App on my iMac (running Big Sur 11.7.10, the latest OS version my iMac is capable of running). I go to the Account tab, click "Sign In", enter my Apple ID (which is a different ID for App Store and iTunes purchases than my 'master' Apple ID) and password, and then nothing happens. I don't receive any error message but I'm not signed in.

I recently replaced the hard drive on my iMac, which requires me to reauthorize the same computer in order to play music that I've purchased from the iTunes store over the years. Since I can't sign in to the iTunes store, I can't reauthorize the computer. When I play a playlist that contains songs I've purchased via iTunes, playback stops and a window pops up saying that the track is not authorized on this computer and would I like to authorize it. When I click 'authorize', I'm prompted to enter my Apple ID/PW. After I do this, I get a message saying "unable to complete this; try again later". This has repeated for days now.

There doesn't seem to be any way to authorize a computer on the Apple account log in website, nor via the System Preferences app within MacOS. It looks like the only way I can do this is to sign in to the iTunes store, which I'm not able to do.

Note that I have no problems signing into the iTunes store on my MacBook Pro, which is running Sequoia. The problems are only on my iMac (which is where my music library lives).