For your symptoms of oily scalp, I would suggest you to clean your hair frequently with a mild shampoo(you can use the ones for oily hair) and don’t apply any conditioner on the hair. Also you can shampoo twice during each wash and apply shampoo on the scalp without piling the hair on top.
Regarding the hair fall, the causes of hair fall could be due to stress, fungal infections, alopecia aereata, hormonal imbalance(overproduction of dihydrotestosterone and thyroid hormones irregularities),malnutrition, telogen effluvium and sebum plugs on skin.
I would suggest you to consult a dermatologist and get these possibilities evaluated. It is very difficult to precisely confirm a diagnosis without examination and investigations and the answer is based on the medical information provided. For exact diagnosis, you are requested to consult your doctor. I sincerely hope that helps. Take care and please do keep me posted on how you are doing.