I wouldn't say "unethical" but I don't think it's a great idea. Although it does reward students who do the study problems, it will disproportionately reward students who can remember line-for-line exactly what they did, over those who instead focused on learning the general concepts.
Also, it only works once. After the first time, it will become known among students that you do this. Lazy students will know that they can get a friend to solve all the problems, borrow their solutions, and memorize them.
As to whether you have the autonomy to do this in a coordinated course: it depends. "Coordinated" covers a wide spectrum of practices. In many cases it is expected that the instructors will cooperate to write a single common exam. Even if not, I think this is something that you should not do without the coordinator's agreement; it's far enough outside the norm that the coordinator may feel that it is not equitable between your students and those in other sections.