I did follow those instructions. They seem irrelevant to my actual issue.
I went to Settings > [my name] > iTunes & App Store and logged out/in. THERE my AppleID shows the NEW email.
In Settings > [my name] > Name, Phone Numbers, Email I see my NEW email as the AppleID.
However, in Settings under my name, there is a line named "Update
Apple ID Settings." When I click on that it informs me that "Some
account services require you to sign in again" then gives me a link to
"Continue." When I hit Continue, a window rolls up entitled Apple ID
Password, and asks me to enter the password for my OLD email, which is
the only email displayed, and which cannot be altered. I entered that
old PW, and was told the verification failed. Verification also failed
when I inputted my new password...
The new password works with my AppleID
when used with my NEW email on everything else including the web from
my computer and other places on my phone.
So, I hope this sheds more light. Thank you for your efforts!