@Jason Hendriks
Excuse me, we're all trying to help - this is not spam if it works. Keep in mind I was using your plugin before I came across this solution. Again this person is having the exact same issue that I was having. They don't need YOUR plugin for this to work.
As you noted in the comments above, the system is connecting to secureserver.net instead of outlook.com. This is most likely happening because the cPanel MX Entry is set to Local rather than Remote. The is GoDaddy's default setting for cPanel hosting.
@tscibilia
I would suggest you disable the Postman SMTP plugin and attempt to change the setting I'm referring to by logging into your cPanel. If you'd like contact GoDaddy and ask them about this and they will show you. It takes two minutes. Once that's done, it should work because now your MX records will determine delivery.
Here's Google's article for reference on this topic: https://support.google.com/a/answer/55299?hl=en
Caveat: I'm assuming that your website is yourdomain.com and you're trying to send an email to address@yourdomain.com via a PHP form.