Can Essential Checkout change shipping prices or add a fee? (Essential Checkout app)

No. Delivery rules work with the shipping options Shopify already shows at checkout. They can hide, rename or reorder those options, but they can't change a price or create a new rate or fee.

To charge some customers and not others, create both rates in Shopify first (Settings → Shipping and delivery). Then add a delivery rule that hides the rate a customer shouldn't get. For example, hide the free rate for certain postal codes so those customers see only the paid rate. See Delivery Method Customizations on Essential Checkout.