Every time I see an itchy scalp/dandruff question I just want to pull out my tea tree oil and be like, "This is the oil of life." Which is to say, I use to have an itchy, flaking scalp, and incorporating a tea tree oil based shampoo really helps get rid of it for me. I use Tea Tree Dr. Bronners Magic Soap, but it isn't very moisturizing. Shea Moisture has a tea tree shampoo, and other shampoo companies do too. Tea Tree oil is really light, like the oil itself practically feels like water, so it probably won't make you more oily. I can't be sure though, because I have a dry scalp. All that said, I know it doesn't work for everyone.
You could also try dry shampoos on second and third day hair. I'm not sure on exact brands, but if you look up dry shampoo in the search bar maybe you can find a thread about them.