In order to change your e-mail address, you also have to fill in the new password, even though it's marked as "optional." You cannot go to that web page, leave the new password area blank, and put in only a new e-mail. If you try it, clicking on [Save Changes] is unresponsive (nothing happens). So, I would say that the new password is not "optional" on that page (as currently programmed).
What you can do, however, is use your current password as the new password value (that is, change it to itself).