Is my battery to blame, or a hardware malfunction?

Hey there. So I have an issue with Keeping the phone on. I guess I should break it down into charging vs pure Battery as I can sometimes do a poor job of explaining things properly. Note: the screen glue is worn off so I can easily remove the screen with my fingers (if I wanted to), and most of the time w/case on one of the bottom corners will be sticking up.

Turning on when charging: Turns on like normal, but 8/10x phone will instantly turn off as soon as it reaches the lock screen. But since it's plugged in, instantly shows battery life screen w/no indication (supposedly) of battery charge loss.

After unplugging when turned on: Will shut off w/o warning at various battery levels. Started off only below 20%, now could have as high as 85% when it shuts off. When you plug it in to charge it shows no signs of a low battery

Turning on when using Pure Battery: Only gets to Galaxy logo then turns off.