Why am I not seeing my new Apple ID on my device when signing in ?

If you have changed your ID or password, have you followed the steps in this article on all devices/computers? Make sure you turn off Find My Phone before signing out of the old ID. Once signed out and Find My Phone is off, then sign in with the new ID, then turn Find My Phone back on.

What to do after you change your Apple ID or password - Apple Support.  *. 

You are sign in here .  .  .  with your Apple ID & Password.

Your Apple ID is the personal account you use with all Apple products and services. It includes the email address and password you use to sign in, as well as the contact, payment, and security details that you use with Apple.

You can’t merge multiple Apple ID accounts.

👉 Sign in to your Apple ID account page and edit / update your Apple ID details.

https://appleid.apple.com

-> edit your 'new' Apple ID.