My experience is a bit restricted but Darwin does not pose any problems as the three lock gated marinas encourage visiting yachts to use their facilities. Best to contact them directly. Two of the marinas most certainly do encourage live a-boards.
During the dry season when overseas yachts are passing through there are plenty of readily available moorings available or you can go on the pick.
Darwin is a great place to spend a few months, particularly from late April to late September. The joint jumps.
Cairns is a mixed bag. Anchoring near the city requires some care and local knowledge. Try contacting the Cruising Yacht Club adjacent to Norship's yard. The main marina in the city, Marlin Marina is just ridiculously overpriced for yachties as it serves the charter
Yorkies Knob yacht club is about 10 km north of Cairns and also usually has vacant pens for rent, but check.
Nearby is Bluewater
Marina which allows live a-boards, but is only about half full. The reasons for this degree of vacancy should be checked out carefully with the manager and some of the residents. It has unresolved problems.
Further north is Port Douglas which has a good marina and other facilities including river mooring or anchoring. Live a-boards -- no problem generally. A tourist/retirees town.
Cooktown is essentially the last stop north unless you are off to Lizard etc. and there are extremely limited tie up facilities but anchoring is available. No live a board issues ...except for the shallow depths and the crocodiles.
Comments regarding the Australian circumstances should be considered carefully. There is a general feeling that many authorities either see yachties as a source of unlimited money
or are there to be abused. There appears to be little support for the cruising industry and this is reflected in the extreme drop off in numbers of international yachts passing through Darwin or visiting the Queensland coast. However, circumstances vary considerably.
Have you sailed out of Simons Town or Cape Town
? Houts Bay? Mossel Bay? Or Knysna etc. The welcome there is invariably better. Make sure you visit Mauritius on the way here.
As South Africans, you may get a shock when here because labour costs are very much higher in Australia. As is food
or anything that has high labour input. Plus we call traffic control lights "traffic lights" not "robots". We do drive on the same side as well.