The batteries are faulty. The laptop detects full charge by the voltage across the battery. It stands to reason that if you remove the power supply and the voltage drops below a nominal supply voltage then batteries are faulty.

Run a battery report.  From a command prompt with admin privilege, type

powercfg /batteryreport

What are the current and design capacities of the battery?

Done.  Thank you for the help.  

Design Capacity: 41,440 mWh

Full Charge Capacity:  32,960 mWh

I left the battery out all night and put it back in today.  When I did that, the power light at the corner came on, but the laptop would not actually power up.  If I pressed the button, the light would go on and off, but the laptop would not turn on.

Does that look like a healthy battery?

Is there any way to figure out if it is the motherboard without replacing the battery or the adapter?  A replacement battery is around $25, which isn't bad, but if it doesn't work, I'm just out that money.  An adapter is $15-$20.  And if it is the motherboard, I'm not comfortable digging around in the computer's innards to take care of it.  I'm decent with software, and usually pretty good on diagnostics, but a laptop's insides are more complicated than I want to risk on a computer I need to be using every day.

But it gets very annoying to lose everything because the adapter cord comes unplugged from the computer due to a little shift on the desk nearby.  And I'd love to be able to move around with my laptop again.

Does anyone have any ideas?  Is there a trustworthy diagnostic tool for a motherboard, or is a process of elimination kind of thing (if a new battery and a new adapter don't work, then it must be the motherboard)?

I suppose I will have to take it to our local repair guy when I get a chance to.  It might be a while, but I will try to post with what he says about it, since I'm not getting any answers here but lots of people have looked at this topic.  Maybe it will help someone else.  

Even a local shop is not going to be able to definitively pinpoint the cause without replacing the battery, which in turn is about half the value of the system at this point.  And most shops that are reputable will charge a diagnosis fee, which will probably exceed the value of the system at this point.

So I bought a replacement battery off Amazon for less than $20.  I didn't think it would help, but I tried.

A new battery did not fix the problem.  It behaves exactly the same with the new battery.  So, no, not a faulty battery.

Basically just letting others know that this problem doesn't always mean a bad battery.  I don't know what the problem is, but it isn't the battery.  I wish I could find more answers on what this could be, but whatever it is, it probably is beyond my ability to fix without risking doing damage to a laptop I actually do need.  

Haven't had a chance to take it to our local repair guy, who is awesome and did a deep interior clean of my other laptop when it stopped working because of dust and such inside the system without charging me for anything.  Just asked for a good review (without mentioning the free service he gave).  And he's been there for years, so eventually I'll get around to taking this system in to be looked at, though I suspect that whatever is wrong with it would mean a repair so expensive I might as well buy a new laptop.

So that's where things stand.  Laptop still doesn't work without being plugged in.  A new battery did not help.  Feel it is likely a hardware issue inside that I am not qualified to fix, but I can't be sure.

@Shamrock​  Did you find out what the issue was, am having exactly same issue. and a new battery costs over 100GBP

I'm having the same issue. My laptop would shut off when I unplugged, or even plugged in, the charger. Now my laptop won't start at all... Only had this thing for about 3 years and definitely won't be going with Dell again after this..