Thank you!
"Click and hold down the mouse button on the Messages icon in the Dock. The popup menu should list the conversation/person with the unread message."
This was the first "fix" that actually worked. After I was able to see which messages were unread, I opened them, which cleared the "unread" badge. " I could then either keep the messages or delete them.
I never saw this fix posted anywhere else, even after searching several different websites and discussion boards. Kudos to LynchGraphics!