After much changing, resetting and failed attempts, I hit on the possibility that the Sent Folder itself was corrupt or broken in some way.
I deleted the existing Sent Folder, rebooted just to make sure it was gone. Then created a new folder I called "Was Sent." ( other possible names were Scent or Cent )
Changed the pointer in Account Settings>Copies & Folders>Other to show Was Sent on Local Folders. And harrah, son of a gun, viola and whew...it's now working properly and leaving a copy in the Was Sent Folder.
[he said as shaking up a new batch of martinis]
Thanks for all the suggestions and help.
