Email Relay Hacked. How do I send email without going through server relay?

Hey guys. Thanks in advance for the help.

My GoDaddy VPS keeps getting hacked by spam scripts in my WordPress install. I have 110 sites up on one server and it’s so annoying. GoDaddy charges me $150 each time they remove malicious script and it’s getting expensive.

So, I need a series of solutions.
1. How do I send email out of the WordPress website via Gravity Forms and generic contact forms, without going through my server.
2. How do I stop the bad scripts.

I want this form question to focus on the first solution as it enables me to keep the emails flowing while I work through the spammer issue over time.

I thought WP Mail SMTP would solve it for me. It does not. Maybe I set it up wrong. Emails go out through my server to the Gmail account, but they seems to be getting stopped since the mail relay on my server is already over the limit for the day. Then I get angry customers, hence the reason this needs to be my first item to fix.

So the BIG QUESTION is, “How do I send email out of my website so that they never touch or go through the website server email relays?”

Thanks guys. I run a non-profit and help lots of people with projects and programs. All my websites are philanthropic. I really need this fixed fast.

Cheers.
Sincerely,

Tony Locke