I spent HOURS at the local Genius Bar trying to get this issue fixed. Hours more on several phone sessions with senior tech advisors at Apple.
But the guy on the phone actually pointed me in the direction of resolving the issue myself.
My problem was that I had several different contact cards for myself, each for a separate purpose, but they all had my Apple ID on them (both email and phone) and 2 of them had my secondary Apple ID that I use for App Store and iTunes Store purchases.
I modified those cards so that only one of them contained my main Apple messaging ID phone number and email. That information was removed from ALL other cards. then I made sure that the same contact cards on my phone were identical to those on my Mac.
Restart, restart.
Problem solved: AirDrop and HandOff now both work as they should in both directions.