Okay, having run several tests, it appears that the welcome e-mail is going to come from the teacher assigned to the course, if there is one.
If there is no teacher assigned to the course, the welcome e-mail comes from the original site administrator, regardless of whether another administrator has been assigned to the specific course as "Administrator".
The Course Manager settings only determine the names displayed in the course listings and have no effect on the welcome e-mail.
So...if the individual who left was the original administrator for the site, it appears that editing that user's profile is the only option. The challenge here is, subsequent users assigned "admin" rights cannot edit the original admin user's profile.
It would appear at that point that your only options are:
1) Assign teachers to all courses so the welcome messages come from them. (A workaround which leaves the main problem in place.)
2) Log in as the admin user you are trying to modify, if possible. You can then alter the profile.
2.1) If you do not know that user's username, but have access to the site's config.php file, you can find the username there. (Probably won't help unless you happen to have the password, which is unlikely if you don't already have the username.)
2.2) If you do not have the password for the user's account, but have access to the underlying
database, you can alter the password directly in the database (as well as the user's name, etc.). There are several posts in the forums about how to reset the admin's password - a little tricky, since it's encrypted, but the posts explain it - you'll have to search, I don't have a link handy...
2.3) If you haven't access to the database, nor the ability to log in as admin...well it seems to me you're kind of stuck with option 1 unless you can contact the admin who left and solicit his/her help.
Good luck!
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