Since it sometimes takes several hours for the system to send email notifications out,
I don't suppose there is a way for the Root Admin to verify that another admin's announcement had that option checked is there?
I would not want to contribute to redundancy of emails by checking that box again in the Post Announcement - Edit - page.
T.I.A.

There is not. Unfortunately, you just have to have that conversation with your admins to make sure they are checking the box.