Okay I found out some more information about the emails that are being sent out.
There was one point where someone set up Mac Mail to temporarily retrieve the business emails for a photography business, and put in an automated responder with an out of town notice. Something about the responder didn't work ~ it started responding to old or deleted emails, something like that. So the rule was deleted, the photography email account was deleted and all of the emails were deleted from Mac Mail. But something unusual is happening ~ it looks like that out of office response is being sent out now as my Outlook emails are being imported into Mac Mail. These appear to be the emails being sent out now. The sent mail is not appearing in my Mac Mail Sent folder, but it is appearing in my Outlook Sent folder.
Is there any way to clear Mac Mail out completely, from the Library folders or somewhere that is not in the application itself? Or can I reinstall it somehow to clear out this old glich that seems to be still stuck? How do I get rid of these back end issues?