Hi,
We moved to Sydney just over a year ago (family of four), I also wanted a house rather than an apartment. We used the same websites you've mentioned to search for a property, and then as we got to know the agents went to them direct. During the time we were looking, I think I saw two properties that had private signs up rather than being with an agent. And both properties were more holiday lets than long term rental properties.
We ended up moving to the South Eastern Suburbs - you'll need to pay $700 - $1,000 for a good quality house, but could pay much more if you've got very specific criteria.
We looked at about 40 properties before finding what we wanted. My main piece of advice is to allow time to look around properties, and be prepared to extend your search area. Property viewings aren't usually by appointment here (like in the UK), you have to wait until the property open day. Then, if you like it, you need to get your application in quick because there will be other interest on the day. We created an application pack (cover letter explaining employment situations, references etc) that we had ready to go as soon as we found somewhere we liked.
Good luck!
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