在WordPress中使用AJAX发送电子邮件和更新表

I'm currently making a WordPress plugin where I want to be able to send e-mail to people from a list. I have made the form (id: campaign), and all checkboxes seems to work. The form submit to itself with a query string, so in the same file I have a if-then that check if this query string has some data.

I have made an admin page with a sub form underneath that lists all e-mail addresses:

I want the script to send out e-mails to each e-mail that has been selected, updating the MySQL table that the mail has been sent, and showing a counter on the page (i.e. '#/15 e-mails sent'). I know how to make this script in plain PHP, but my lack of knowledge with AJAX is the problem.

So this is what I want:

1. The form sending data to itself (same PHP file) - this is working.
2. Starting the AJAX script based on the selected checkboxes
3. Sending out e-mails and updating the SQL table based on a timer (to avoid spam)
4. Showing real-time counter

This is what I have so far:

    <div id="feedback"></div>
    <?php


    add_action('wp_ajax_sendPromo', 'sendPromo');
    add_action('wp_ajax_nopriv_sendPromo', 'sendPromo'); // not really needed

    ?>

    <script type="text/javascript">
    jQuery('#campaign').submit(ajaxSubmit);

    function ajaxSubmit(){

        var campaign = jQuery(this).serialize();

        jQuery.ajax({
            type:"POST",
            url: "/wp-admin/admin-ajax.php",
            data: campaign,
            success:function(data){
                jQuery("#feedback").html(data);
            }
        });

        return false;
    }
    </script>
    <?php

    function sendPromo(){

    global $wpdb;

    $recepients = $_GET['recepients_email']; // array of checkboxes
    for($i=0; $i < count($recepients); $i++){
        mail($recepients[$i], $subject, $message, $headers); // mail is being sent
        if($wpdb->update('wp_ap_promo',array(
        'sent'=>1
        ), array( 'email' => '" . $recepients[$i] . "' ))===FALSE){

            echo "Error";

        }
        else {
            echo "Successfully sent to '".$recepients[$i]. "', row ID is ".$wpdb->update_id;

        }
    }
    die();
}

One of the problems seems to be that the JS isn't being run because the page is reloading (it is meant to reload, because I want the user to see a plain page with the send counter.