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Use the below code snippet to show the shipping name instead of billing name in the order number column in the admin dashboard. Add the code in the active theme functions.php file.
add_filter( 'woocommerce_admin_order_buyer_name', 'woocommerce_admin_order_buyer_name', 10, 2 );function woocommerce_admin_order_buyer_name($buyer, $order) {
$buyer = trim(sprintf(_x('%1$s %2$s', 'full name', 'woocommerce'), $order->get_shipping_first_name(), $order->get_shipping_last_name()));
return $buyer;
}