You were out of status when you quit I believe. That means that you probably won't be allowed to change your employer in the US. They would ask for your last pay stubs and if there is a gap they can make you go back. I believe you can enter 10 days before you are set to start. I'd get a lawyer for all of this if I were you.
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Your E3 status ended when you quit you job. Yes, you need to go back to Australia.
Your employer needs to retain an experienced immigration lawyer, whether myself or one of my colleagues, to review all the facts, advise them, and handle the case.
J Charles Ferrari Eng & Nishimura 213.622.2255 The statement above is general in nature and does not constitute legal advice, as not all the facts are known. You should retain an attorney to review all the facts specific to your case in order to receive advise specific to your case. The statement above does not create an attorney/client relationship. Answers on Avvo can only be general ones, as specific answers would require knowledge of all the facts. As such, they may or may not apply to the question.