Open/Close Cart Drawer
Try clicking the cart button on your store. It should open Unicart and not the cart page or theme's default cart drawer/mini-cart.
Cart Items Count in Header
If your theme shows a cart items count in header or next to Cart Icon, make sure it is being updated when you add or remove items from your cart.
Cart Total Price in Header
If your theme shows cart total in header or next to Cart Icon, make sure it is being updated when you add or remove items from your cart.
Add To Cart
Try adding the items from your shop's Home, Products, Collections, Search or any other pages on your store. It should add the product in cart only once and should automatically open the cart drawer with the added product.
Facebook Pixel
Test if your Facebook Pixels are firing correctly. They should not fire twice when you add a product into the cart.
Translations
If you have enabled multiple languages for Cart Drawer, make sure they are working.
Multicurrency
If your store sells in multiple currencies, make sure the drawer shows correct currency.