Problem Solved, was fairly simple.
I tried a couple of suggestions, one I thought may have worked but did not, doing, "I FORGOT MY APPLE ID PASSWORD, under the ICLOUD preference, and then changed it AGAIN, slightly, changing the "A" on the end to a"B" but that did not work.
Then I Googled this:
"Getting a Pop up after booting Up that my Mac can't connect to iCloud because of a problem with the Mac Sign in Addy, and I am on MacOs Sierra??
And found scrolling to a second page this, did the following AND it did not happen again upon RESTARTING, a second AND THIRD time>>>WHEW:
"Go to the Apple menu and open System Preferences, then choose iCloud. Select to “Sign Out”. Once the Mac has booted back up again, return to theiCloud preference panel, and login to the Apple ID as usual.Oct 23, 2016
When you do that SIGN OUT from ICloud, it gives you the impression that you will have to choose a completely different password for getting into ICloud and that you will lose all your folders & Files, but that is not the case. I did have to choose a NEW PASSWORD, but I chose the same exact one as my MAC ID Password, which it accepted, and then you are given two options, 1) To Save all your ICloud data on, "THIS MAC," or 2) To remove it all. Why would anyone do that, LOL. I chose # 1.
All well now. Did several RESTARTS, and no longer get that, "This MAC can not connect to ICloud," Popup. Like I always tell me who ask TECH questions online, "Google is your BEST FRIEND." :-)
P.S. That, "Library" stuff sounded too risky to me, and also to a good friend of mine who is a computer Wizard, and ditched Windows for MACs awhile ago. I knew there had to be a better, way :-)