So I am a dual citizen currently residing in Australia, but I need to go home to America pretty much ASAP (2 days) due to family circumstances. I entered Australia on my now expired Australia passport, would it be okay for me to leave Australia on my valid American passport to go to America? I would be staying there for quite some time and would then update my Australian passport? I still have my Australian passport to show I entered legally, it is just not valid to fly on anymore.
Countries generally prefer you to leave on the same passport you entered the country in. Of course this isn't always possible (citizenship change, for example).
The Home Affairs website notes for dual citizens:
A dual citizen is a person who is a citizen of 2 or more countries.
You become a dual citizen if you:
are an Australian citizen, are granted citizenship of another country and don't lose your Australia citizenship, or remain a citizen of another country and become an Australian citizen
Australia allows a person to have dual citizenship.
Of course, now you just know they recognise it.
And so they note, regardless, if you have an Australian citizenship:
Australian citizens entering and leaving Australia
Australian citizens should use their Australian passport to enter and leave Australia.
The Australian Border Force has more information on crossing the border.
If you have a passport from another country, you can use that after you leave Australia.
A passport is the best way to show that you are an Australian citizen. You might still be able to enter Australia if you are an Australian citizen without an Australian passport, but it will be more difficult. The airline might also stop you from boarding a plane to Australia.
Smart Traveller has more information about travelling as a dual citizen.