You need to do the following to sign off the old Apple ID from iCloud on your devices:
1. Go to Manage Your Apple ID, sign in with your current Apple ID and password, and change the Apple ID back to the old email address. This is temporary. Do NOT verify it, do NOT change the password: Apple - My Apple ID
2. Go to Settings>iCloud on your devices, and scroll to the bottom to Sign Out of the old iCloud account. When it asks for the password, enter your current Apple ID password. Follow the prompts to complete the sign out process
3. Go back to Manage Your Apple ID and sign in with the old ID and current password and change the ID back to the new email address. You may need to verify it again
4. Go back to Settings>iCloud on your device and sign on with your new ID
Cheers,
GB