Hello,
With the Storefront I see the exactly the same issue with the cart subtotal while changing the shipping method https://gyazo.com/1d656cef345ec969a226657d9127550e
It means that our theme is not the reason of the problem and we can’t fix anything using theme files. Check the plugins that you use for the WooCommerce shipping for example. I suppose one of them is the reason of the problem and contact plugin author.
Regards