After you've logged out your account in settings -> app store, create and sign in on the new account. Bear in mind apps are tied to the accounts the purchases were made with.
That's also because apps are DRM protected, your current apps belonging to the old ID is permanently protected under that account. Through updates, iTunes will also continue to write DRM to the old ID you had because those apps were once DRMed under the old ID, therefore requiring authorisation with that old Apple ID which is why it keeps "coming back" when it really didn't.
You can only update apps on the accounts you've signed in to.