Updating icloud email

Currently I have a Gmail account as my Apple ID, and an icloud.com email address as a "Reachable at" contact. Note, I have the iCloud email address unchecked on my Messages and Facetime on my iPhone.

I want to delete my current icloud email address and replace it with a new icloud email address (I recently got married and changed my name).

When I log onto https://appleid.apple.com/account/manage --> Account Edit , there is no way for me to delete the iCloud email. It says "This is an alias of your Apple ID email address"

When I log onto https://www.icloud.com/mail/ with my Gmail account (my Apple ID), I am automatically logged in with the original iCloud email and cannot delete or change it there either.