Thank you, Bruce! We fixed it.
Working through the remove/reboot and WMUtil steps above, I stumbled on the source of the problem. The duplicate-message e-mail address had found its way into one of the other accounts. So, checking the other account
also collected messages from the problem account.
WMail popped up a sign-in box during in the wee hours of Wednesday morning, as it is wont to do every once in awhile. Usually, we just cancel and life goes on. But, late at night, the box triggered the respose that used to be required when we had DSL --
signing in with the e-mail account that suddenly got sick.
The result of that sign-in was to put the problem-account e-mail into a second account. Both accounts were checking the same 'post office box' and then sorting the incoming to the same inbox.
Correcting the second account fixed the problem.
Thanks to Bruce for good advice and patient coaching.