Are you able to send us your website link and admin access so we could check why it doesn’t work for you? And please perform the following test also: activate WooCommerce and set search for posts, will it work? And specify what header layout do you use.
We see that in general, your search works for blog posts only. The only problem is that AJAX results show products (and you don’t have products at all). So we simply disabled AJAX search since it is not compatible with blog posts and dedicated only for WooCommerce products.