Does Shopify have an option to show out-of-stock items but disable the "add to cart" button?

You have two options if you want to hide the "Add to Cart button" for your sold-out product. Either you can disable "Continue selling when out of stock", or you can change the theme's source code.

To disable "Continue selling when out of stock", from your Shopify admin, head to settings > Products > Scroll down to inventory and check the "Continue selling when out of stock" option.

If this doesn't work, or you want to change the code, you can wrap the "Add to Cart" button code with an {% if %} statement. Keep in mind, if you're not familiar with changing code then it's best to outsource, as incorrect changes can be hazardous to your website.