My PHPmailer script can't send email to a valid account on the same domain if the domain is hosted externally. How can I fix this?

This is a direct result of your local website host trying to deliver mail to what it considers a "local" address since it's on the same domain, when in reality that email exists on an external server, for example, Google Apps. Your mail server (such as postfix or sendmail) should have a setting to disable this "shortcut".

If you are using CPanel, for example, the setting can be found here: cPanel -> MX Entry -> Email Routing -> Remote Mail Exchanger