I can’t get my Apple ID to work on Messages or FaceTime. The old messages from another Apple ID are coming up

See If you get an error when trying to activate iMessage or FaceTime  https://support.apple.com/en-us/HT201422

Get a call from Apple support to help solve your problem using this link https://getsupport.apple.com/?caller=kbase&PGF=PGF63005&category_id=SC0245&symptom_id=23362

Apple ID Support https://support.apple.com/apple-id

Apple ID account security for country-specific and region-specific help

 https://support.apple.com/en-us/HT204169

Have you signed the old ID out of Settings/Messages and Settings/FaceTime, then restarted the device, and then signed in with the desired Apple ID?