You only need to pay tax in Australia, if you are a considered a tax resident in Australia. Since you are leaving the country, and are going to live in the USA, you will probably be not an Australian resident anymore, so you don't need to pay taxes in Australia. The government has a site where you can check your tax residency.
There is also another page, which will describe what you need to do if you are leaving Australia, like doing your taxes for the current tax year. There might be also capital gains taxes you need to pay for some of your Australian assets if you intend to leave the country.
Also you will still need to lodge tax returns for any Australian income you might have:
If you become a foreign resident, you will only need to lodge a tax return if you have Australian income (ignoring any income from which non-resident withholding tax has been deducted, such as bank interest and unfranked dividends).