Can't log out of iCloud on iOS7?

You have to delete the account, then sign back in with the ID you wish to use. Deleting the account only deletes if from your phone, not from iCloud. Doing this will not effect anyone else still using the account.

However, when I choose Delete Account it asks for my password based on the Apple ID I no longer have. So there is no way to even delete iCloud

If the old ID is an earlier version of your current ID, temporarily recreate the old ID by going to https://appleid.apple.com, click Manage my Apple ID and sign in with your current iCloud ID. Click edit next to the primary email account, change it back to your old email address and save the change. Then edit the name of the account to change it back to your old email address. You can now use your current password to turn off Find My iDevice, even though it prompts you for the password for your old account ID. Then save any photo stream photos that you wish to keep to your camera roll. When finished go to Settings>iCloud, tap Delete Account and choose Delete from My iDevice when prompted (your iCloud data will still be in iCloud). Next, go back to https://appleid.apple.com and change your primary email address and iCloud ID name back to the way it was. Now you can go to Settings>iCloud and sign in with your current iCloud ID and password.

THANK YOU!!!

Have the iPhone 6+ and was unable to delete account because of the password. Have been freaking out but did what you said and it worked!

Thanks!!

I tried to edit my current mail to the former one .. But it shows this email id is not available