Don't know the full details of who to contact, hopefully someone else can help out with the Govt. department to talk to; also, this is just my thoughts on the subject, but hope it helps in some way.
The reality of your situation sounds like you're actually working as a full time employee, and should therefore be entitled to all the things that employers are supposed to provide for their staff, like super, paid leave, sick leave, and the rest of it.
If you were a casual, you wouldn't be working on the roster, and should be able to say when you are and are not available to work.
If you were a contractor, you should have an ABN and provide your own personal and public liability insurance, and have a specific contract as to what service you will provide, and so would need to be paid appropriately to cover your extra costs as opposed to an employee.