I have got the correct new cost displaying but the calculations of the total cost still be made with the old cost of the method.
I need the new cost be the actual cost for calculations in the cart and check out

add_filter( 'woocommerce_cart_shipping_method_full_label', 'custom_shipping_method_full_label', 10, 2 );
function custom_shipping_method_full_label( $label, $method ) {
    if ( !in_array( $method->get_method_id(), array( 'free_shipping', 'local_pickup' ), true ) && 0 < $method->cost ) {
        $label    = $method->get_label(); // Get shipping label without cost
        $new_cost = $method->cost - get_total_subsidies( WC()->cart->get_cart() );

        if ( $new_cost > 0 ) {
            $method->cost = $new_cost;
            $label .= ': ' . wc_price( $new_cost );
        } else {
            $method->cost = 0;
            $label .= ': ' .' <span class="free-cost">' . __('Free!', 'woocommerce').'</span>';
        }
    }
    return $label;
}