are you using a script to send or software/ IEM?there are lots of options. I've used sendgrid and would recommend it. there's also amazon ses, mailchimp, etc.
Can you set it to mail through smtp?I send e-mail by Pommo. It is super easy and flexible. Every user can unsubscribe by special generated link and also I can add my own lists of e-mail to subscribers.
It has got an option, where you can set how many e-mail should be sent every second and how long should be a spaces between XX numbers of e-mails. By this option, I can easily run it on shared website hosting.