Here's a couple of links to the two hospitals you've mentioned:
Since you're wanting sponsorship, I'm thinking the RAH would be your best bet.
If you're going to live in Nairn, you might also consider the Murray Bridge Hospital: www.mbsmb.sa.gov.au
Also, here's a couple of links for real estate, rentals etc:
I hope you might find these useful.
I trained at the RAH but that was many many many moons ago!
I have worked there several years ago when doing agency nursing. Naturally, with the passing of time, things had changed. However, I still found it pleasant to work there and the staff were friendly and supportive. As they SHOULD be!
I'm a bit biased where the RAH is concerned, I loved my time there and have wonderful memories of having lived in the nurses quarters and lifelong friendships made. I can tell you it is one of Australia's leading public hospitals though. and in saying this, I'm not being biased!
That's just fact!
Wishing you all the very best as you navigate your way "Down Under".