Alan,
For best results your ListMail "Bounce to email" or (bounce disabled) "Administrator Email" should be on the same domain as ListMail is installed on. While not always required, the "Send From Email" for your list may also need to be on your ListMail domain. Also not always required, the bounce and from addresses may need to exist in your control panel as an email forwarder or account.
If you aren't using the SMTP AUTH feature, try enabling it using an active mailbox for authentication.
Regards