I just tested on my boards(6.7.1) where I remove profile update rights from a test user, and change the signature. I was unable to duplicate the behavior.

Trying to visit the edit profile link nets the error (FYI
Sorry, your profile has been specifically locked by the board administrators. This means that you may not make any changes to your registered profile information. )

Are you sure you are not viewing a cached page with the old signature?? A way to test that would be to edit a post on the page, and submit. That will invalidate the cache for that page, and reload avatars, signatures, etc.