Signing Out old Apple/iCloud ID

If you forgot to sign out of iCloud (and had Find My iPhone or Find My Mac turned on) before you changed your Apple ID email address, then you need to do the following to get the old iCloud ID signed out of your iOS device or your Mac:

First go to Apple - My Apple ID and turn off Two-Step Verification or Two-Factor Authentication if you have either one of those options turned on for your Apple ID. This is a temporary step to facilitate getting the old iCloud ID signed out. If you have 3rd party apps using iCloud, then these may get signed out as well, and would require you to sign those apps back in with an App-specific password.

On an iOS device that is running an iOS version earlier than 10.3:

1. Go to Settings>iCloud, and if Find My iPhone is turned on, tap it to turn it off. Type in the current password for your Apple account

2. Then click on Delete Account at the bottom of the screen and follow the prompts selecting the options you wish to complete the sign-out

On your Mac:

1. Click on the >System Preferences>iCloud, and if Find My Mac is turned checked, click it to deselect it. Type in the current passcode for your Apple account

2. Once Find My Mac has been disabled, click on Sign Out and follow the prompts selecting the options you wish to complete the sign-out

Note: If you have an iOS device that is running iOS 10.3 or later: you won't have to do anything. With iOS 10.3 or later, as soon as you verify the new address, the Apple ID/iCloud address will automatically change on your iOS device (Settings>Your Name), In other words, if you have a device running iOS 10.3 or later, you should not see this situation at all.

Don't Forget!!!: Once you have completed the sign out, and successfully gotten the updated ID signed in, be sure to go back to your Apple account page and set Two-Factor Authentication, or Two-Step Verification back up again

Note: If you set up an entirely new Apple account, then you have to supply the password for that other Apple account in order to sign out. If you cannot remember the password, then go to If you forgot your Apple ID password - Apple Support and follow the instructions that apply to the type of security setup you have for that Apple ID. If you are using Security Questions as your setup, and you don't remember the answers to your Security Questions, contact Apple Support to have them reset ACCOUNT SECURITY CONTACT

Future Tips: Whenever you change your Apple ID, before doing anything, sign out of the following on all of your devices as well as your Mac:

iCloud

iTunes & App Store

Messages

FaceTime

iCloud

Setting up a brand new Apple ID is setting it up by creating a new account. Doing that allows you to start completely fresh with no information from your previous account on it. This, of course, means that you will no longer have access to any media purchased under the Apple ID that was used on that other Account.

So, unless you have to open a brand new Apple account for personal or other reasons, it is always best to keep the account you have, and just change the Apple ID/iTunes/iCloud email that is associated with that Apple Account. That way, all of that account's history remains with you - All of your purchased Music, Movies, TV Shows, or other media are forever tied to the Apple Account that they were purchased under, so if you set up a brand new Account, then you are starting all over from scratch.