As most others have already mentioned, Wyoming is a great place to start.... but with that said, don't drive through Montana on your way to Wyoming and not at least consider it. Montana has some very good antelop hunting and it's a bit closer to home for you. There are areas that require a draw and there are areas that you can buy over the counter.
One aspect of Wyoming that I don't think has been mentioned, is that you can combine it with a meat hunt as well. I don't know if you're after the meat, but you can buy doe antelope and doe deer tags over the counter at a very nominal fee. Go after that 16 inch buck but take a couple of does for the freezer. When antelope are cared for properly, they are absolutely delicious.
Don't you wish they would introduce a good herd of the goats in your state of Washington? I think it's been tried before but failed. I'd guess the habitat near the tri-cities would make a great place to try again. In fact, I think I read somewhere that it has recently been tried. I don't know why they wouldn't thrive on the East side of the state.
Wherever you go, good luck. What part of Washington do you do your guiding? Must be near Wenatchee if you grew apples.