Oily hair, dandruff and dht all play a role in hair loss. Sebum can clog hair follicles and thus can be a contributing factor to thinning and hair fall to hair loss.
I would recommend a shampoo with otc ketoconazole like Regenepure DR shampoo. Do give it some time, as initially you might see more hair fall but eventually in a month or so reduce hair fall, less flakes, and healthy scalp, especially with oily hair as because of keto it has a drying effect. For complete list of ingredients visit their website.
Regarding regrowth, I would recommend 5% minoxidil which is FDA approved for male pattern baldness. Regenepure too have the spray for that as their main line of products are for hair loss, hair care, maintenance and hair growth.
Disclaimer: I have been using their products for myself and recently started importing their products for selling in my country as I was very happy with their results.