My advice is to contact beef / sheef companies (Like AACo, Stanbroke, etc) HR dept and ask who to send a resume to for station flying. I did some time on a station, ended up with an extra 400 hours in my logbook after a year and had some fantastic experiences to boot!
As for time to go on the drive, I understand it to be round about now actually! Just after the wet season is a good time, as you don't want to be green and have to put up with a NT wet season!
Good luck and hang in there! Once you're in, it only gets better!