You have 2 problems:
- Your header section HTML structure is modified:
Yours:
From the demo store, same theme version:
See how you theme has div, while demo store has header-component? Because of this, some theme Javascript does not work and your mobile menu is not initialized.
- Even with this fixed, you have some CSS added by the Giraffly Trust badges app (https://apps.shopify.com/trust-badge-by-giraffly) which overrides some styles used by your theme code to show drawers for menu and cart. Have you noticed that when you add to cart, the screen gets darker but no popup is shown, while it should look like this:
Someone needs to have a look at your theme header, also worth contacting the App support and complain about incompatibility.