While you still can reset your iPhone without Find My or iTunes, the behaviors you describe sounds like a digitizer failure. When an iPhone's digitizer doesn't work properly it typically can cause taps in unexpected places. Your passcode also couldn't have changed as that is stored in a very secure place in your iPhone (The Secure Enclave), which makes it so the passcode cannot be modified or erased in any way. Just to add some info, without Find My activated it only prevents using the easier "Forgot Passcode?" option. Without iTunes, (which is present on computers), you can visit the Genius Bar to assist in resetting your iPhone and to have your iPhone serviced, likely the screen replaced.
Genius Bar Reservation and Apple Support Options - Apple