Hi Suzanne
I personally think you are going to be struggling The rental market is really tight here at the moment, with a lot of competition for unfurnished longer term properties, we always recommend that people coming here consider taking a furnished rental for up to 3 months, to give you enough time to get a feel of the suburbs you like and to secure a longer term rental in this competitive market. Yes it can be done in 4 weeks, but with Xmas in there when not a lot happens, I wouldn't like to chance it. Can you extend your stay at the Freo proeprty? If not let me know as we have a couple of properties that may suit for a further couple of weeks.
When applying for rentals Include everything that you can, with the aim of making it easy for the RE agent to choose you over others
Good luck
Tracey