Joe,
The only places you can hunt for goats without entering a draw is in Alaska and the western Canadian provinces of British Columbia, Yukon and the NWT. All these hunts require a guide for nonresidents. Unless you are hunting in Alaska with a resident that is a close family member of yours. Guided goat hunts can run from $5,000 to over $10,000. Depending on where you want to go.
Alaska is the only state where U.S. citizens can hunt sheep without entering into a drawing for a tag. If you are not an Alaskan resident you must hunt with a guide or have a close family member that is a resident that you will be hunting with. A guided Dall sheep hunt can run from around $8,500 on up to over twice that much. Depending on where you go Alaska or Canada, Alaska being the least expensive.