I have hesitated to respond to the suggestions that have been submitted, especially by AlWeir. Thank you for trying, Al. The reason for my hesitance has been the problem went away on its own. First 1 of the 5 cleared up. Then a day later another one cleared up. At this point I was elated. Then, on the next day, 2 more came back. Now I figured that since the remaining one was not used at all, just a vestige of the past, I deleted it from macOS mail and all 4 are continuing to work properly.
In my search for a solution the only thing I could find as a possible cause was that the accounts were not using the correct outgoing servers. I typed in the correct ones but it didn't seem to make any difference. But then, the next day they were OK. I can't figure out the real cause of the problem, so I hope the next "Bug Update" that seems to be coming out soon will kill more bugs than it creates.
Bob