Jen,
Yes, you can keep them on the NSW licensing system. All species of reptiles (excluding crocodiles, sea-turtles and sea-snakes) are permitted to be kept in NSW through the DEC (aka NPWS) licensing system.
There are 2 classes of licence. Class 1 (basic) and Class 2 (which also has different endorsement levels for certain species of venomous snakes).
The NPWS website should still have a file which lists what class of licence you require for a large number of species. This list does not include every Australian species of reptile as it was compiled before the decision was made (I think this was 2004) that allowed every species to be kept.
For information on what Class a particular species is, you can contact DEC. I am pretty sure you'll find that Thorny Devils are Class 2.
Now, after going through all of that, if you can find (and adequately house and feed) some, all the luck to you!!
Simon Archibald