Salvo's sometimes have places. I know there is womens accom at Campbelltown.
if you can't get into some temp housing try a caravan park. They aren't super cheap but you can usually get one with a double and 2 singles for around $35-40 a night. Well that is the price I paid last time I stayed in Launceston when I was visiting from Hobart.
Anna have a look at the NSW Department of Housing website under Emergency Temporary Accomodation...
I'll post the link but in case the mods need to remove it you should be able to find the page.
Sadly, the way things are you usually have to be living in your car to be classed as an emergency. The dept of housing has 5-10 year waiting lists.
I can't believe no one can help you out in whats a pretty dire situation?! surely family or friends could just let you crash on the couch or a spare mattress even?
Theres an agency here called "parents without partners" which may be able to organise respite accomm if theres something similar in sydney.
Anna, going to PM you
xox
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