Hi there!
I am Belgian, 35 years old, I have more than 10 years of real professional experience as a network engineer, I have worked with Nestle, Ozone, Lan10 and many small companies and ISP.
Looks like hostgator does have a throttling policy on outgoing emails, you can check on:
www dot hostgator dot com slash mailpolicy
I guess it depends of the hosting contract you have with them, however, with an access to the server I may be able to check if it is their policy or some configuration problem on your server and fix it, I would recommend postfix or exim but this is up to you.
There is a chance it is not possible with their servers, if this is the case, I can of course recommend other solutions for you.
I would be glad to work with you, and solve this problem, I am available right now for the next 20 hours, you can contact me here or on skype: protojob, anyhow, I wish you good luck!