Removing iCloud account without password

A couple of months back I changed the email address of my Apple ID, because I no longer had access to the email address I used. Everything was working great, I can sign in to iCloud and apple services using the updated email as ID, but when I changed my password yesterday I ran into some issues.

It turns out my mac was still logged into iCloud using the old email, and after changing the password I was logged out of iCloud on my Mac. That's of course expected, but the problem is it won't accept either my old or my new password. There is no Apple ID connected to my old email address anymore.

I think the only way to change the email used to log into iCloud on the Mac is to remove the iCloud account and add a new one. But when I go to do that I get this prompt, and I can't proceed without a working iCloud password.

Is the only good way to get around this to start with a fresh os x install, login with my updated Apple ID, or is there another version to change your Apple ID?