I ended up contacting Apple Support, but representative had no answers and then set up an appointment with Tier 2. Between my initial chat with Apple and the scheduled Tier 2 call, a message popped up on one of the users of my MacBook Air that attention to iCloud was needed, sending me to System Preferences. I don't recall the exact wording, but it was something about logging into iCloud for all services to be available. The iCloud login popped up, and I logged in. However, I had already re-logged-in to all my devices last week after the Apple ID got locked. But after this last night, ever since responding to that message, now every device/user I have opened up gives the message, "You have added iMessage and FaceTime to a new device" and each message pop-up would list a different device. And now Messages works on my two computers.
What I don't understand is why the re-login last week didn't solve this, and why it took Apple's system a whole week to suggest a login that would affect iMessage and FaceTime. It seems that all that I did was to log into iCloud AGAIN, despite having logged back in last week after the Apple ID got locked.