Yeah, your options are pretty limited. If you're really keen, save up and go on a guided hunt in Canada or Alaska. Goat hunts are considerably cheaper than sheep, when I was looking last year, most were in the $7-10,000 range. With the strong Aussie dollar and the weak US economy, you might be able to find a deal.
Some states have raffle hunts as well, I know Oregon and Washington do. Basically you buy raffle tickets and hope for the best. Unlike putting in for draw tags, you don't need any upfront money, just the ticket price. But, the odds are lower. Can't win if you don't play though and somebody has to win! The raffles are usually around May, but check the date in what ever state you're looking at.
I must say, thar and chamois are a pretty good substitute!