Sydney - very gay, and a lot of fun (I live there!). Do not miss it. It is a pity you are not travelling Nov- Mar, as these are our summer months, and everyone is out partying. Many will be hibernating in September. But you will still have a great time.
Melbourne - also good, much more suited to winter than Sydney. Also decent gay scene.
Brisbane - will be warmer, but I don't think there is a great scene there (although I haven't really looked for one when I have been there).
Adelaide - never been there but (and I mean no disrespect here) I can't see any reason to go there.
Agree that you should go further north for water & sun. Don't think that you'll find much gay life though, certainly not compared to Sydney.