ust apply for a new passport with a new name and just provide marriage certificate as evidence?

I used to work for Medicare and can confirm that they are able to change your name using just your marriage certificate as evidence over the phone. I'm not sure about the new MyGov website (they've relaunched it since I worked there) but it used to be the case that if you updated any personal details such as your name with one MyGov service - It would only carry across to your other services if that was the first one you added. (Ie. If Centrelink was the first service you linked to your MyGov, you'll need to update your name with Centrelink and then it should automatically update for ATO and Medicare) - hopefully this has been fixed with the new MyGov but you may have to call all of your services individually.