Hi @Sinso ,
Clemency from Zapiet here - we create apps for Shopify. I spotted your question and wanted to help out!
It sounds like you may have both Shopify Local delivery and Shipping enabled. From your description “Under shipping, I have added a list of postcodes…” it sounds like you’ve added this list of postcodes into your Shopify settings > Shipping & delivery > Local delivery settings. If customers from all over Australia can check out though, you may also have Shipping zones set up.
If you do not offer shipping, then you may wish to disable your shipping zones. It sounds like you have an Australian shipping zone set up, so once that is disabled, only customers in your Local delivery area will be able to checkout.
As Shopify does not provide shipping zone information in advance to customers, any customers who try to check out in other parts of the country will get to checkout, and see an error message, and may get frustrated and contact you.
You may be interested in an app that can screen your customers pre-checkout. Zapiet - Pickup + Delivery is an app that is added to your cart page where customers can select from the checkout methods you offer (e.g. local delivery, pickup, or shipping) and add their address to see if they are eligible for local delivery. If they are eligible, you will also have the option of a calendar, so customers can choose their delivery date/time, opening up your business to future orders as well 
We offer a 14-day free trial, live support and set-up calls to help you to configure the app to your needs so you can give it a try