Laptop runs out of battery while powered off

So I didn't mention that the battery is not bad, and that it's capacity is fine when the laptop is on, just not when it's off. Sorry about that. But I found the culprit- I saw the laptop at night, running. The screen wasn't on, and neither was the SDD indicator, but the fans and the backlit keyboard were on. I had to shut it off using the physical power button. It was on that same blank screen problem as I had mentioned in the original post. Maybe it is the display? I have no idea.

Now I'm trying to figure out how to stop it from turning itself on from a completely powered off state, besides removing the battery (which isn't so easy to remove).

Well, I tried turning off Hibernation completely and running a malware scan. I also made sure Maintenance isn't allowed to wake the computer for scheduled maintenance.

I will try the display driver if none of that works- which I will know by tomorrow.

Also, I should mention that when it shuts down, it shuts down completely. It just has been randomly turning back on after a while. And only once- when I caught it running and shut it back down, it didn't turn back on for the rest of the night. When it was running like that, or when it wakes up from sleep running like that (with the screen off), the backlight of the LCD and keyboard are both on.

So I was wrong, since I'm not currently the user of the laptop- My father is. The problem is actually this:

The computer isn't turning itself on, it's actually not turning off (Sometimes)!

I saw my dad turning it off, and while the screen turned off, I noticed the fans were still going.  I pressed a key, and sure enough the backlight on the keyboard turned on.  But everything else seemed to be off.  The Caps Lock key's light did not illuminate when I pressed it as it normally would.  I had to force it to shut down using the power button and holding it for a few seconds, until the backlight turned off, and the fans did as well.
So this makes more sense than it turning itself on from a completely powered off state, but I am still clueless.

And yet, after making sure it was off last night and checking to make sure it was STILL off a couple of hours later, my dad tried to turn it on the next morning and the battery was completely dead. Not even enough power to tell him to plug it before it gets to Windows, like it normally did with this issue. Plugged it in and it turned on. I am really not understanding this laptop...I blame it on Dell replacing it's motherboard with a bad one, but it's not under warranty now.

Sorry for all the trouble...Any suggestions?

Good idea, I will try that as soon as the laptop is available again. The clean driver installs I will do as well. It was a clean Windows install, though.

So, we aren't using the laptop as often so it was hard to tell, but the issue is still there. I believe it is either the new Solid State Drive, since I don't remember this issue occurring before we moved (Which is when I installed the SSD), or the motherboard replacement since Dell definitely gave me a worse one than it originally had. I'm going to go with the SSD, since the laptop is out of warranty. I'll contact Samsung and I'll see what they can do.

Thanks for your efforts! I really appreciate it.