If you are pulling your hair why the hell any php form on your site hosted on GoDaddy is not working, you are on the right place.

The reason is because GoDaddy added some kind of “security” feature that will not allow you to send e-mails from accounts that doesn’t exist in the system (CPanel).

So for example if your form is sending emails from behalf of wordpress@site.com it is not going to work.

What you need to do is to go the CPanel and create a freakin mail account from which you want to send the mails.
I mean WTF GoDaddy? Can’t you at least mention this on your freaking site?