Hello Alexandra,
Thank you for the details of the issue you are experiencing with iCloud on your devices after updating your Apple ID. I recommend the following steps:
Symptoms
iOS 7 may ask for the password to your previous Apple ID when you sign out of iCloud.
Resolution
1. Use your current Apple ID password
Trying using your current Apple ID password. If that doesn't work, continue to the next step.
2. Sign out and back in to iMessage or FaceTime
If you are currently signed in to iMessage or FaceTime
If you are not signed in to iMessage or FaceTime
3. Change your Apple ID temporarily
If signing out and back in to iMessage or FaceTime didn't work, try these steps:
- Change your Apple ID to the Apple ID you used previously. You shouldn't need to verify the email address.
- Tap Settings > iCloud and try to sign out.
- Change your Apple ID to the new email address that you want to use. You'll need to verify the email address.
- Tap Settings > iCloud and sign in with the new Apple ID.
Find the full article here:
iOS 7: If you're asked for the password to your previous Apple ID when signing out of iCloud
http://support.apple.com/kb/TS5223
Thank you for using Apple Support Communities.
Best,
Sheila M.