Just as chemotherapy drugs cause hair lossby damaging the fast-growing cells of the hair follicles, they can also damage the skin cells of the scalp. And of course the scalp is now less protected thanks to the hair loss. Sensitivity is a common problem as a result; in addition many cancer patients report itching or flaking. It sounds like the symptoms you are experiencing are very severe, however, and I suggest you call your doctor to discuss them. It's possible you are having an allergic reaction to one of the medications you are on, such as a steroid, and your doctor needs to know.
If you haven't already, you might find it helps to shave your head. Many cancer patients find this more comfortable than dealing with dry, thinning hair and itchy stubble. Also, make sure you protect your scalp with a hat or sunscreen at all times.
Doctors generally advise that you continue to shampoo, but if this is causing you so much discomfort you could try washing less often. If you haven't already, you need to switch to a non-irritating shampoo free of perfumes or dyes. Pat dry gently rather than rubbing with a towel. If this isn't enough, try rubbing an over-the-counter hydrocortisone cream on your scalp.
Doctors and nurses often recommend replacing your normal cotton or linen pillowcase with a satin one, or one made from smooth cotton sateen, as even the texture of regular cotton cloth can irritate a sensitive scalp. Read more suggestions here.