i can not remove my old work email address

Welcome to the Apple Community.

In order to change your Apple ID or password for your iCloud account, you need to sign out of the account from all your devices first. (Settings ➔ Apple ID(Top) ➔ iCloud, scroll down and hit “Sign Out” from an iOS device or System Preferences ➔ iCloud and click the “sign out” button from a computer)

Note: You can only change a non-Apple address, you cannot change an iCloud, Me or Mac address.

Note: You may also need to amend your details for the iTunes & App stores, iBooks, Messages, and FaceTime.

Once you have signed out of all your devices, you then need to change your details with Apple. Start here and edit your Apple ID and/or password. Once you have completed your changes you can sign in again on each of your devices.

Providing you are simply updating your existing details and not changing to another account, when you sign out from your account, all the data that is synced with iCloud will be removed from the device (but not from iCloud) but will be synced back to your device when you log in with your new details.