I'm pretty much in the same boat: Godaddy won't send my Newsletters at all. To anyone, not even me. Following that thread mentioned in a previous post prompted me to send Godaddy an email to ask for an SMTP relay increase, to which they responded:
The SMTP relay limit for our shared hosting accounts is 1000 per day. Unfortunately, this limit cannot be increased. If you need more than 1000 relays per day, you will need to move to our Virtual Dedicated or Dedicated Server plan. These plans also have a 1000 relay limit per day but that limit can be increased.

More information on Virtual Dedicated Hosting can be found here.

More information on Dedicated Hosting can be found here.

Please let us know if there is anything further that we can do for you. Our service departments and telephone lines are open 24 hours a day, 365 days a year to accommodate your needs anytime.
Well, I don't Need 1000 relays per day... 1000 per week would be fine. I cannot afford to upgrade to Virtual Dedicated Hosting... not yet, anyway.

Just want to send periodic newsletters - fortnightly or monthly - to my customers to show them what's new. Can't understand why this is so hard to do from within Zen-Cart... seems like it should be a basic function.