Maybe your just playing a new sport that forces you to use new muscles!
I haven't seen your swing, but I highly doubt that it is causing the problem, especailly if both wrists are hurting. It could be that your grip is way out of whack, but I think someone would have noticed and pointed it out to you. If your wrist muscles are stiff and sore, try stretching them out before, during, and after every round.
Play through it. It WILL get better. If you want to do something about it now, you could start lifting weights. All you have to do for forearms and wrists is to hold a dumbell in each hand, palms down, then rotate your wrists up and down slowly. Hope I helped.
P.S. Congrats on taking up golf!