If you fail validation you are just filling an errors array and continuing with your script.
// begin: reCAPTCHA - VALIDATE
$resp = recaptcha_check_answer ($privatekey,
$_SERVER["REMOTE_ADDR"],
$_POST["recaptcha_challenge_field"],
$_POST["recaptcha_response_field"]);
if (!$resp->is_valid) {
$errors[] = 'Please enter a valid captcha';
}
// end: reCAPTCHA - VALIDATE
You need to wrap everything after this point in an else
block, otherwise you just fill your $errors
array and happily proceed with the rest of your page. You aren't even using your $errors
array later.