Ohhhhh.... I hope someone can explain this to me so I can preserve my sanity!
To recap: The problem was that when the tablet was plugged in - AND the battery definitely showed that it was nearly FULLY charged, and there was the lightning bolt symbol indicating that it was plugged in and charging - when I would unplug the cord from the tablet, it would shutdown. I have been struggling with this problem for months.
Reading the more recent suggestions, I decided to try @guru's easiest suggestion first: Seeing what would happen by unplugging the power cord from the outlet instead of unplugging it from the tablet. When I unplugged it from the outlet, the tablet successfully switched to battery power with no problem. So, I plugged it back into the outlet, and then the tablet showed that it was charging. THEN, I tried unplugging the cord from the tablet, and..... It successfully switched to battery power. Now when I unplug the cord from the tablet, I no longer have the original (or any) problem.
What is the explanation? What should I be concluding? Am I just being lucky at this time, and next week the problem will return?
After I wrote the above, I happily used the unplugged tablet for the very first time in MONTHS!!!!!! While unplugged, I shutdown the tablet. Then, while still unplugged, I tried restarting it. It simply would not. (When I plugged it back in, then it would restart.) But, while it was unplugged and I was trying repeatedly a few times to power it on, I noticed something I had never noticed before: The screen of the tablet was actually "bowed" such that it was separating from the electronics underneath it inside the tablet - I could actually peer inside and see the guts of the tablet. I do not understand why this happened, the tablet has always been treated very gently. Anyway, I am now thinking that likely there may be some resulting hardware problem(s) inside the tablet itself that is causing the power problem????? (You may be wondering, How could he have not noticed that until now? But, because I could not use the tablet unplugged, I always left the tablet where it was, and moved my chair to it, instead of picking it up, especially since where I had the tablet sitting did not leave a lot of slack in the power cord. So, I never picked it up to handle it or look at it all these months. It just never occurred to me that something like this would happen,)
Now I am thinking, I'd like to simply purchase a new, reliable tablet with Ubuntu Touch from UBports already installed on it, instead of continuing the frustration with this tablet's unreliability. Are there reliable sources for such a tablet that I can examine online?