Hi there!
There is a Facebook page called Adelaide Rental Property you could check out, there's plenty of lease breaks and private landlords advertising direct on there. However, I would think Realestate.com.au would have the most choice.
Many people obtain rentals by just showing their bank statement or job offer, but your application will be getting compared to anyone else's who wants to rent the property and theirs may be more favourable if they have a checkable rental history.
Some rental managers won't accept an application from someone who hasn't seen the property, and you would be unwise to apply for a place you hadn't checked out in any case.
Landlords are keen for tenants to move in as soon as possible so any that are willing to wait for you to arrive would raise some suspicion?
Might be better to arrange a short term rental on arrival then look for somewhere more permanent once you have had chance to look around.
There are members on here who offer short term rentals
Hope that helps a little