yeah, me too.
Everything else on the phone seems to recognize the new apple id (for the most part)
To answer the second question, yes I did. i wasn't required to change it, but felt that it was time to update the password as well.
When I try to enter the password for the old ID, it says the account for the old id doesn't exist.
Settings> Messages> "Apple ID Verification" "Enter the password for 'old apple ID' in Settings.
*taps on settings* *prompts up "Apple ID Password" (still for the old apple ID)
*types in password* "old Apple ID is not an Apple ID"
(okayy? and I close out)
but then whenever I'm at:
Settings> Messages> Send & Receive > "Use you Apple ID for iMessage"
"New/Current Apple ID is Already Signed In"
I try to "Add An Email"
and when I manually type out the current Apple ID and password I'm prompted with "iMessage server is unavailable"