the point i am trying to make is this:
you have the code to validate captcha on the OK page.
you have the code to send the email on the form page.
both of these types of code are processing the form data. They need to both be applied to the forms action page.
right now, you have them separated on different pages.
either change the action of the form to be blank, and add both universal email and the captcha code to the form page. (Prefered method)
or leave the action as is and add both universal email and the captcha to the OK page. (Not preferred method)
If they need to login to get to the contact form anyway, then No, you do not need captcha in place.