iPhone 5 shuts off / loses power, no matter what I try

I got this phone from craigslist recently. It powers on and you can use it, but only for a little while. The phone just shuts off after a few minutes. The battery shows as charged. Sometimes you can watch it fade to black, lol. After it loses power, pressing the power or home buttons does nothing. You can't even power it back on by holding the power button as you would normally. You have to unplug the battery (remove the screen and battery connector covers first obvously), plug it back in, then you can power on the phone. I discovered later that holding both power and home for several seconds will turn it back on.

I figured it might be the battery, but despite trying several different known good iPhone 5 batteries, all were charged, I get the same behavior. I even swapped out the Lightning Connector and Speaker Assembly, thinking that may be it, but that didn't seem to help at all. I also did a factory reset, and a DFU, to no avail.

I am hoping it isn't the motherboard, but it may just be that. Would be a shame because it is a 32 Verizon phone, no icloud lock, and I got it for cheap, thinking it just needed a new screen.

I forgot to mention, the water damage stickers are all white, not red, as red indicates water damage. But even though they are not red, that doesn't mean it ISN'T water damaged...

Thanks