I would only like to use the short description (since it shows directly under name and above price) and then hide the long description. I know I can hide the regular long description using CSS but is there a way to also hide it in the admin?
Description is default woocommerce tab and unfortunately, there is no option to disable it in admin panel. Just don’t add full description for your product, leave the main description field empty.
If you already have main descriptions for your products and want to hide it without editing products you can use WooCommerce snippet to remove specific tabs https://docs.woocommerce.com/document/editing-product-data-tabs/snippet
All the changes should be made in child theme function.php to avoid problem with next theme update.