Well, I hauled my 45 pound MacPro into the Apple Store yesterday...
What they told me there was that it appears as if the problem is related to those using GoDaddy mail; NOT GoDaddy's mail using Office365 (the only mail service you can buy now from GoDaddy) but rather GoDaddy's hosted servers, which are imap.secureserver.net and smtpout.secureserver.net. Apparently the culprit might (I have yet to let it run for days) be the checkbox under mail preferences-->accounts-->mailbox behaviors-->drafts. You need to uncheck (i.e. do not) the box about "store drafts on server".
As explained to me by the Genius (that's what they call them there!) helping me, there needs to be some kind of handshake on this that either the GoDaddy servers aren't doing, or Apple Mail isn't processing properly or both causing a "memory leak" which results in the "out of application memory" error message.
So we unchecked the box on just the GoDaddy server email accounts and we'll see what happens over the next week.
This also might explain my OTHER issue which drove me to Apple Mail in the first place: when I set up my email accounts on Microsoft Outlook (Office 2011 for Mac) for these GoDaddy servers, they worked fine for 20 minutes. Then, they lost the connection to the IMAP server and the only way to get them back was to delete the IMAP mail account on Outlook, and re-install it. That of course, was only good for 20 minutes until THAT lost the connection.
I'm not a fan of GoDaddy and sorry I ever signed up with them and seriously considering moving my domain and mail to someone else.