I spoke with Godaddy and what they told me is that the emails are being sent via PHP and that whenever I trigger he plug in, they see the relay sever being activated to send the email. I also tried this using the SMTP and the settings they gave me and they do not see the relay server being triggered. So they suggest that I go back to PHP mail, even though the emails are not making it through.
They told me that they can not know what is being triggered to be sent so they think that the plug in is triggering the relay server but what is being sent is not making it through.
The end result is that all the test mails are being sent without a problem. The PHP mail is being triggered and being sent. So from their perspective PHP mail is working fine, all the tests are working fine, so there is something in either the newsletter or subscription mails that is creating the problem. So there is nothing else that they can do.
I have tested this with a test newsletter that just has test as the subject and it does not go through.
Further ideas?