I wouldn't write airbnb off completely. We stayed in a lovely house through airbnb, just outside the CBD. Not cheap (insurance company were paying) but very well set up for a holiday stay. We also stayed in an apartment actually in the CBD just off Rundle Street, also through airbnb. We've also used it for accommodation in the UK, elsewhere in SA and Port Douglas. Stayz.com.au is another website worth having a look at. I prefer airbnb these days though as you have the reviews section and if a place is not suitable you can call them and they will find you an alternative place to stay.
Alternatively have a look at the accommodation section on here. @Tamara (Homes Down Under) has several places for rent and may have something suitable for your dates. I believe her places are around the Port Noarlunga area, so not really central for exploring all of the different areas around Adelaide, but a lovely area to be in.