I had an iPhone X that needed back glass and new battery. I traded it in for credit toward a newer phone. Just what I did.

Why is 13 not for you?

As far as I know the part that plugs into the wall should work no matter what. Sounds like it’s a problem with the adapter or the wall outlet itself. Try other outlets and test etc

Are you talking about the part that plugs into the bottom of your iPhone? That is now usb-c. Your iPhone 13 had lightning so that’s what’s different.

You should definitely bring it to Apple, especially these hardware issues. Mine has been generally fine.