One of my fosters is like that. His hair is super oily and skin flaky...but, he also has itching from it. He has to be bathed once/week...sometimes twice; and he gets cytopoint injections every 4 - 6 weeks for the itching. I use this shampoo which really helps:
https://smile.amazon.com/gp/product/...?ie=UTF8&psc=1
Here is a link about sebhorrea in dogs which is what this sound like in yours:
https://vcahospitals.com/know-your-p...orrhea-in-dogs