If I get orders to a restricted tour to Korea, could my wife move back to her hometown on the army's dime?

You can move your family but it will be at your expense. The Army will pay to move your family to the follow on duty station only.

The way HAAP works is that you are offered a location. If you accept the location then that is your guaranteed follow on assignment and you can move your family there ahead of time or not. Or, you can deny haap and you will receive an assignment while you are in Korea

Pretty sure Korea works either of 2 ways.
1) You go and she stays at Hood to then move when your Korea Tour is up to wherever you are assigned.
2) You go and she moves to your post-Korea assignment (which is already figured out) and she'll be there ahead of you.
Hometown stuff is prob not an option under the Army's dime, however. Those two options above should be. They can't just say your dependents are homeless because they assigned you to a restricted area. They will house your family either where you are before Korea or where you'll go after Korea.

Good luck with that year. It was a rough year from me when I did that in Germany. Deployments too.