How to change Apple ID?

From what you post it appears you cannot add an iCloud alias as your Apple ID, you can only use the iCloud email address that is the basis for your iCloud mail account which is the old name. If you do not want that as your Apple ID then you need to set up an external email account or you would have to create a new Apple Account.

Please click on the links and read the documents -->

Change your Apple ID - https://support.apple.com/HT202667 - "Follow these steps to change the email address that you use as your Apple ID."

"After you change your Apple ID email address or password, you might need to update your sign-in information with any Apple services you're using" Click on this link to read "What to do after [or before] you change your Apple ID email address or password" --> https://support.apple.com/HT204071

- You may wish to disable Settings > Find My before changing your information, change the information, then reactivate it with the new information.

There's special instructions if you want to change your Apple ID to an @icloud.com address.