Icloud email (Apple ID) verify response "email address not available"

You are using old mail address as an Apple ID to sign from iCloud.com say example1@me.com see the article iCloud: About your @icloud.com, @me.com, and @mac.com email addresses - Apple Support

Better change the Apple ID say example2@icloud.com or say example3@gmail.com you had already created Change your Apple ID - Apple Support

The account is set up with 2FA Two-factor authentication for Apple ID - Apple Support

Get a verification code and sign in with two-factor authentication - Apple Support

lisafromsealbeach,

We appreciate those additional screenshots and details. 

To confirm, are you already signed in with this Apple ID on your Mac or another device? If so, are you receiving a verification email in the Mail app on any of your email accounts associated with your Apple ID account? 

Set up iCloud for Mail on all your devices

If not, then we would recommend contacting our Apple ID Support to get the best assistance with this issue going forward. 

Get Support

Cheers!

The apple id is signed in Mac system preferences > you know the password is correct , if Apple ID is incorrect or its identity is zero on the server this link will help https://iforgot.apple.com/appleid#!&section=appleid

A green check mark will appear if account is verified , your screen shot clear shows that Apple ID address is incorrect stating use another Apple ID .

Take suggestions from senior apple care advisor Contact Apple for support and service - Apple Support

This article Set up iCloud for Mail on all your devices - Apple Support

is helpful for adding a new Apple ID address in device it could be Mac / iPhone and its not relevant to the problem of primary Apple ID ending with me.com ( change your Apple ID you are using latest version of Mac and in it this Apple ID is outdated and not compatible with 2FA account ) .

Hello, lisafromsealbeach.

From the screenshot you're sharing, this shows that you are attempting to set up a new email address, not sign into one. If you recently attempted to set up a new address, have you checked your email for a confirmation from Apple? In that email, you should see a link requesting that you confirm the address. When you click that, it should take you to a page where it shows the confirmation is successful.

After that process is successful, when you want to log in to iCloud, you can go to the iCloud login page and sign in there.

Please let us know more about the exact process you are taking to sign in and if these steps are completed successfully. Take care!

If Apple ID or its password is changed from Mac it has to be always update in synced devices What to do after you change your Apple ID or password - Apple Support

lisafromsealbeach,

You're very welcome. We hope you enjoy the rest of your day.

Stay safe and well!