Also I wanted to let others know that this could be legitimate and by that I mean the spam emails... I've cleaned up a few hacked sites where the hacker gained enough info and uploaded shell scripts to upload email scripts so if you were hacked at any point in time and did not clean it up thoroughly (110% lol) then your site IS sending out those emails because there's something quite nasty on your sever doing so sadly.
*This can happen if you were ever hacked OR another account was hacked lets say on a shared server etc, if you just moved to a new host it is also possible the old host did not catch onto the behavior and the new host did, the problem could have been long-existing and you're just now finding out because one host is on par more than the other.
Here's the gist of it, in your email programs or windows live mail for example you can enable and view email (message) headers, that there can tell you IF they originate from your site, they can make the email say it's from your same exact email, it will say from "youremailaddress@yoursite.com" but in the header it will give specifics and you can tell by that if its in fact from your site originally or not.
https://support.google.com/mail/answer/29436?hl=en
So with all that said, if I sum it up, you may not be sending out the emails however your site could still be sending them if compromised.