Changing Apple ID without signing out

Assuming your operating system is up to date, you should be able to enter your new password to sign out, once you have signed out completely, sign back in with your new password.

As I say I believe this should work for you, but if not here is the advice I used to give with earlier software...

If you mean that you are being asked for a password to a different Apple ID to your current Apple ID and that ID is a previous version of your current ID, not an entirely different one.

This feature has been introduced to make stolen phones useless to those that have stolen them.

However it can also arise when the user has changed their Apple ID details with Apple and not made the same changes to their iCloud account/Find My Phone on their device before upgrading to iOS 7, or if you restore from a previous back up made before you changed your details and some other circumstances.

The only solution is to change your Apple ID back to its previous state with Apple at My Apple ID using your current password, you don’t need access to this address if it’s previously been used with your Apple ID, nor do you need to verify it, once you have saved these details enter the password as requested on your device and then turn off "find my phone" and delete the account from your device. It may take a short while to remove the account.

You should then change your Apple ID back to its current state, save it once again and then log back in using your current Apple ID. Finally, turn "find my phone" back on once again.

This article provides more information about Activation Lock.