I'm very grateful for your help, but for the sake of full disclosure I'm not a total Mac newbie. This was my first computer with its 90MHZ processor and a whopping 1 GB HD!Okay, so your best bet would be an external storage drive or to upgrade your iCloud plan (I have the 200 gig plan for like 3 bucks per month). My iCloud stores all my email SERVER settings (which means that when I tick the mail boxes in "Internet Accounts" under system settings, I don't have to keep entering any smtp server info. I just have to log in with my u/p when prompted.
If you use IMAP or Push email, then your emails should be stored on your email server until you delete them using your client. So when you log in, the mail.app client will begin downloading all those messages from the server, and they won't be deleted from the server unless you delete them from the client. The exception to this is if you use POP email. As I noted in a previous message, that will remove the email messages from the email server as it downloads them to your client, but few people use POP these days -- I'd check to see what email protocol you've been using (i.e., if it's Gmail, it's almost certainly not POP). The payoff of what I'm saying here is that deleting your partitions, after logging out of all your iCloud/iMessage stuff, will NOT permanently delete your mail messages: they'll likely still be sitting on your mail server. I'd double check this stuff, though, just to be sure you don't accidentally delete important info., as I am not actually at your computer to verify what your settings are or aren't.

Power Macintosh 7200 - Wikipedia
You have taught me something that I didn't know; that iCloud saves my settings from Internet Accounts, although I keep all mine in a TextEdit document, along with passwords etc.
I've just updated to 11.2.2 and nothing's changed.
Of prime importance to me is my mail that goes back to 1995, and is still of great use to me. Just today I was emailing with a client and I was able to show him the invoice he paid back in 2010 to prove that my rates had not increased since then.
All of that mail is in folders in Mail.app, not on servers.
I run my own mail servers on my domain with various accounts and forwarders so I'm pretty much converse with how things work.
All my accounts are POP, including my Gmail accounts and I delete mail frequently from the servers, specially Gmail, so everything stays on my computer, and my Time Machine backups.
I guess that after a clean install that you recommend I should be able to restore all that mail from TM without restoring whatever may be causing this bug?
While it may seem to many like a minor glitch it's a PITA for me. Today I was exchanging emails every few minutes with a client while doing other stuff. I have Mail launched but hidden, and without those new mail sounds and alerts it's hard to know if I've received new mail without looking at the dock icon, which doesn't always bounce either.
I have custom sounds for many recipients so without looking I should know who's emailing me. That's a luxury I don't have any more either.
Like this 😉