With very greasy hair it would be well worth an appointment with a Trichologist who would look at the condition of your hair and scalp to see if there is any underlying cause for over production of sebum. Over production of sebum causes greasy hair! As the hair is falling out, you most definitely need to see a Trichologist.
The best treatment for greasy hair is to wash it every day, avoid conditioner other than to the ends and use a specially formulated shampoo that wonít exacerbate the appearance of the hair. Look for a product that contains silicone and oils. ...More >
Please donít shave it off! Get some professional advice from a Trichologist, see Institute of Trichologists for your nearest qualified Trichologist.
Hope that helps,