Birth control has been an effective form of treatment of polycystic ovariona disease/syndrome. Which is why it has been adivsed to yo as well.
I would also give the go ahead for the shot. Do not worry about hirsutism
(excessive hair growth
in women), this is usually rarely associated with this form of birth control. And if you do experience it, discontinuation of the shots will get your body back to its pre-administration state.
Best of luck and god bless.