Must I change the admin user email address while changing the general email address?

The email address that emails such as new user notification or password resets come “from” is not the same as the site’s admin email. While it’s possible to have the actual email address be the same for both of these, they are completely unrelated to each other.

In WP’s Settings > General, the “Administration Email Address” is the email address used to administer the site. This is saved in the WP db options table (wp_options) as “admin_email”.

This value should not be confused with the “from” email address that emails going through wp_mail() will come from (or appear to be coming from). These are two entirely different things.

If you want to change the address that emails come from, you need to do one of two possible things:

1. Filter the “from” address for wp_mail() using the wp_mail_from and wp_mail_from_name filters.
2. Configure wp_mail() to send through an authenticated SMTP account or install a plugin to do it for you.

Update:
It sounds like you might already be using one of the two above methods to change the “from” address. You’ll need to update that accordingly if you’re changing the email address.