Hi, @adammahon
Greetings, and thanks for posting.
Stopping the sale of sold-out products should not require adjusting the theme liquid. When creating your products, if you want to prevent customers from purchasing products that are sold out, you will need to track the products inventory, and make sure to select ‘Continue selling when out of stock.’
If you have this selected for each of your products, and customers are still being able to add the products to the cart, it could be your coding that has caused the issue. I recommend setting testing on a fresh theme. This means a theme that is freshly downloaded to your theme library and has not had any customizations done to it.
In the meantime, it looks like your store is password protected. If you can provide the password I would love to take a look and see what additional feedback I can provide.