Hi MG, my XH and I had the same arrangement when we first separated, I didn't get the payment from centrelink after a while we ended up doing a week on week off arrangement, so one week I stayed with a friend while he stayed in the house with the kids the next he stayed with a friend while I stayed in the house with the kids, I think then O started to get a payment (I was working part time) but getting the part payment from centrelink helped with me finding my own place. But was definitely too hard to get anything while still under the same roof.
Was a very hard time mentally, emotionally and financially but you'll get there in the end.
Best of luck

ADDIT: Just saw post by tiggerandpooh, my X was very uncooperative and was refusing or holding up any paperwork centrelink or otherwise so that wasn't really an option for me, but I hope you can work it out that way