Yeah, sounds like the battery might have swollen and bent the screen. Happened on my 13 Pro a few days ago—luckily, it didn't damage my screen, but it definitely can. A damaged battery like the one you described could have put pressure on the display, causing internal damage, especially if it was punctured and heating up.
Replacing the screen is a good next step since you're already seeing display issues. If that doesn't fix it, there could be damage to the connectors or even the logic board from the previous issues. When you install the new screen, check the connectors carefully for any signs of burn marks or bent pins. Hopefully, it's just the screen that needs replacing!