Are connected to the your account - which is also your iCloud account? Your name and connection should be the first thing you see right at the top of settings. If you open the sub-menu, it will show just below your name which Apple ID (which email) you are connected to.
imessage, facetime, icloud, and find my phone all sign-out when you change your apple password. There could be others that I don’t use routinely. Go through the settings for each of those apps or launch them and it should prompt you for the updated password.
A warning- once you enable two-factor authentication you can not disable it. I tried to look up a procedure to do it without reasonable success. The only one I found said to deactivate the phone for 2 weeks and it would reset. D’OH!