My understanding is that you can't really change the names of your biological parents on your birth certificate. A birth certificate is an historical document that is critical to knowledge retention in terms of genealogy, so I don't know if it's possible to change the name of who gave birth to you, even if someone else came to be your psychological parent.
What you can is look into adoption. Being over 18, you don't need your parents' consent to be adopted, but you do need to file through the Court. I'm not across the process for New South Wales, you can probably call your local registry for information, but hopefully this gives you some direction to start.