What Hook/Action is performed when a field in checkout form is changed in WooCommerce

You will need to use ajax to set values in WooCommerce session variables first. Once your values are set in wc()->session, then you can access them anywhere within the WooCommerce hooks or actions.

See below code example:

/*Ajax Call Back*/
add_action( 'wp_ajax_my_action', 'my_action_callback' );

$shipping_charges;
function my_action_callback() {
    $shipping_charges =  $_POST['shipping_charges'];
    WC()->session->set("th_shipping_charges", $shipping_charges );
}

/*Ajax code to get values from the form and post to WordPress backend*/

add_action( 'wp_footer', 'my_footer_code' );

function my_footer_code() { 
$html="
    jQuery("#th_sub_area").change(function(){
    var shipping_charges;
        var selected_sub_area = jQuery("#th_sub_area").find(":selected").val();
        if(selected_sub_area="DHA'){
               var shipping_charges= 150; 
        }

       / In front end of WordPress we have to define ajaxurl /  
       var ajaxurl = "'.admin_url('admin-ajax.php').'";

       var data = {
         "action": "my_action",
       "shipping_charges" : shipping_charges,
                };

                jQuery.post(ajaxurl, data, function(response) {
                    jQuery("body").trigger("update_checkout");
                });
    });
';
echo $html;
}