<div class="panel">
  <h2>Search Videos</h2>
  <div id="my-search">
   <form role="search" method="get" id="searchform" action="http://myurl.com/" >
    <input type="text" value="" name="s" id="s" />
    <input type="submit" id="searchsubmit" value="Search" />
   </form>
  </div>
</div>

add_action('wp_ajax_wpa5000_search', 'wpa5000_search');
add_action('wp_ajax_nopriv_wpa5000_search', 'wpa5000_search');
function wpa5000_search(){
  global $wp_query;
  $search = $_POST['search_val'];
  $args = array(
    's' => $search,
    'posts_per_page' => 10,
    'cat' => 12,
    'post_type' => array( 'post','fod_videos'  )
  );
  $wp_query = new WP_Query( $args );

  get_template_part( 'video-search-results' );

  exit;
}

add_action( 'wp_enqueue_scripts', 'wpa56343_scripts', 100 );
function wpa56343_scripts() {
 wp_enqueue_script(
    'wpa56343_script',
    get_template_directory_uri() . '/libs/search.js?ver=1.0',
    array( 'jquery' ),
    null,
    false
 );
 wp_localize_script(
    'wpa56343_script',
    'WPaAjax',
    array(
        'ajaxurl' => admin_url( 'admin-ajax.php' )
    )
 );
}

$(document).ready(function($){
 $('#searchsubmit').click(function(e){
    var $panel = $(this).closest(".panel");
    $panel.empty();
    e.preventDefault();
    var search_val=$("#s").val();
    $.post(
        WPaAjax.ajaxurl,
        {
            action : 'wpa5000_search',
            search_val : search_val
        },
        function( response ) {
            $panel.append( response );
        }
    );
 });
});

<?php
 while ( $wp_query->have_posts() ) : $wp_query->the_post(); ?>
   //STUFF
<?php endwhile; ?>

<input type="hidden" name="post_type" value="fod_videos" />