My pup is almost 4 months she also takes a shower once a week along with my 1 year yorkie. Yes the can get dry skin, Marley had this problem.
My suggestion is to buy a really good shampoo and conditioner that made huge difference in Marley's hair, plus I don't put shampoo on his back(where he was getting dry) only every other week just their face paws and private parts.
i hoped this helped I know is difficult to wait more than one week
