My best friend rents a unit with her 2 little girls in Sutherland and loves it. Her girls go to childcare 2 days a week and there is a park not to far from where they live.
There are many other suburbs in the shire you could live in which are cheaper and still nice area's. I have lived all over the shire before and was born & raised in Bundeena (which is across the water from Cronulla)
Places like Loftus, Kirrawee, Gymea etc are just a quick train ride to Cronulla and in the cheaper rental areas. I agree that you could be applying with loads of others, but that doesn't mean you won't get it. I have many single mum friends who are on benefits that live in rentals in the shire.
I hope you find something real soon.