Contact form won't send emails to any email I put in as a recipient. I don't want to connect form to my server

Question

I am working on a secure contact form and am having a hell of a time, what I am about to ask may be confusing but I will try to explain it as best I can.

I need my contact form to securely send emails to which ever email I put in the php code but it will only send emails that are linked up with my server and I don't want that at all. (ex. it will send to kdawg@kdawg.com. It won't send to kdawg@hotmail.com, gmail.com you name it.) Is there anyway I can fix this in my contact form of is there any other way or direction someone can point me in? I want to avoid mailto: because all the reading I've been doing says it's not secure at all. Any help here would be most appreciated. Thanks in advance.

FORM

<form action="post.php" method="post">
    <div class="leftSide">
        <input type="text" name="name" id="name" required="true" placeholder="Name">
        <input type="text" name="email" id="email" required="true" placeholder="Email">
        <input type="text" name="subject" id="subject" required="true" placeholder="Subject">
    </div>
    <div class="rightSide">
        <textarea name="message" id="message" required="true" placeholder="Message"></textarea> 
    </div>
    <div class="centerRecaptcha">
        <div id="RecaptchaField2" class="g-recaptcha" data-sitekey=""></div>
    </div>

    input type="submit" value="Send Message" id="contactBtn">
</form>

PHP

<?php

// Clean up the input values
foreach($_POST as $key => $value) {
  if(ini_get('magic_quotes_gpc'))
    $_POST[$key] = stripslashes($_POST[$key]);

  $_POST[$key] = htmlspecialchars(strip_tags($_POST[$key]));
}

// Assign the input values to variables for easy reference
$name = $_POST["name"];
$email = $_POST["email"];
$sub = $_POST["subject"];
$message = $_POST["message"];

// Test input values for errors
$errors = array();
if(strlen($name) < 2) {
  if(!$name) {
    $errors[] = "You must enter a name.";
  } else {
    $errors[] = "Name must be at least 2 characters.";
  }
}
if(!$email) {
  $errors[] = "You must enter an email.";
} else if(!validEmail($email)) {
  $errors[] = "You must enter a valid email.";
}
if(strlen($message) < 10) {
  if(!$message) {
    $errors[] = "You must enter a message.";
  } else {
    $errors[] = "Message must be at least 10 characters.";
  }
}

if($errors) {
  // Output errors and die with a failure message
  $errortext = "";
  foreach($errors as $error) {
    $errortext .= "<li>".$error."</li>";
  }
  die("<span class='failure'>The following errors occured:<ul>". $errortext ."</ul></span>");
}

// Send the email
$to = "email@gmail.com";
$subject = "$sub";
$message = "Name: $name \r\r\n $message";
$headers = "From: $email";

mail($to, $subject, $message, $headers);

// Die with a success message
header("Location: index.html");

// A function that checks to see if
// an email is valid
function validEmail($email)
{
   $isValid = true;
   $atIndex = strrpos($email, "@");
   if (is_bool($atIndex) && !$atIndex)
   {
      $isValid = false;
   }
   else
   {
      $domain = substr($email, $atIndex+1);
      $local = substr($email, 0, $atIndex);
      $localLen = strlen($local);
      $domainLen = strlen($domain);
      if ($localLen < 1 || $localLen > 64)
      {
         // local part length exceeded
         $isValid = false;
      }
      else if ($domainLen < 1 || $domainLen > 255)
      {
         // domain part length exceeded
         $isValid = false;
      }
      else if ($local[0] == '.' || $local[$localLen-1] == '.')
      {
         // local part starts or ends with '.'
         $isValid = false;
      }
      else if (preg_match('/\\.\\./', $local))
      {
         // local part has two consecutive dots
         $isValid = false;
      }
      else if (!preg_match('/^[A-Za-z0-9\\-\\.]+$/', $domain))
      {
         // character not valid in domain part
         $isValid = false;
      }
      else if (preg_match('/\\.\\./', $domain))
      {
         // domain part has two consecutive dots
         $isValid = false;
      }
      else if(!preg_match('/^(\\\\.|[A-Za-z0-9!#%&`_=\\/$\'*+?^{}|~.-])+$/',
                 str_replace("\\\\","",$local)))
      {
         // character not valid in local part unless
         // local part is quoted
         if (!preg_match('/^"(\\\\"|[^"])+"$/',
             str_replace("\\\\","",$local)))
         {
            $isValid = false;
         }
      }
      if ($isValid && !(checkdnsrr($domain,"MX") || checkdnsrr($domain,"A")))
      {
         // domain not found in DNS
         $isValid = false;
      }
   }
   return $isValid;
}

?>

If there is anything I need to clarify just let me know I will do my best.