I have had success with a simple hack to the forms. I rename one of the fields (usually the email address) and then add a fake hidden field named "email". If the form gets submitted with something in the fake field the back-end code ignores the form submit.

  <input type="text" name="email" style="display:none;">

  <label>Your email address</label>
  <input type="text" name="fax">

The back-end looks for the real email address in the "fax" field, and silently ignores the form submit if it sees anything in the "email" field.

I don't remember where I saw this trick a few years ago, I didn't think of it myself. It has worked pretty well on multiple sites that have exposed "Contact us" forms.