I am an Australian living in New Zealand and was recently granted New Zealand citizenship. However I only have an Australian passport. Can I travel freely between Australia and New Zealand on my Australian passport?
You'll be aware of the reciprocal arrangements between Australia and New Zealand that allow for essentially unrestricted travel between the two countries for their citizens, so in practical terms, yes, you can travel on your Australian passport.
However, if you do so you'll be admitted to New Zealand as an Australian citizen. With the current Australian stance on admitting NZ citizens it is possible that the NZ government may change arrangements for Australian citizens in the future.
You have two options if you wish to be admitted to New Zealand as a New Zealand citizen: apply for an endorsement in your Australian passport (details here), or apply for a New Zealand passport (details here).