Hi Kevin,
Yeesh, sounds like an ugly problem. Have you verified the AC adapter is putting out the correct voltage for your laptop? That would be my first step to check that with a voltmeter. After that I'd open up the case and measure the voltage on the DC input jack's pins to verify the power is actually getting to the motherboard.
If everything is good up to that point, then I'd seriously start to consider whether a new battery might be called for. Yeah, it's only three years old, but depending on usage the battery could well be starting to fail. That's really about all that's left at this point that's fixable without board-level repairs, so that's what I'd try myself.
The batteries aren't particularly cheap, but they're not outrageously expensive either; the ones I saw were running from about $50-60 USD. Here's one of the first examples I came across.
[link|https://batteryclerk.com/products/lenovo-yoga-c740-yoga-c740-15-yoga-c740-15iml-replacement-battery|Lenovo Yoga C740 YOGA C740-15 Yoga C740-15IML Replacement Battery: BatteryClerk.com|new_window=true]
iFixit unfortunately doesn't have a guide available for replacing the battery, but given that you've stated you've unplugged the battery I assume you know how to go about replacing it.
Good luck with it; let us know what happens!