I always paid my rent upfront - that meant there was some overlap in a two week period when I moved but I was reimbursed later from the real estate.
I'm not up on NSW law but seems common sense you're not obliged to pay rent once you leave a premises - a premises the landlord has demanded you vacate.
That's two bites of the cherry.
If you're not up to date with rent... If it was me I would pay the Dept of Housing rent...and maybe pay back the private rent when you can.
That to me seems a reasonable response to an unreasonable demand.
If in doubt check with the tenants union.
D