Ah, this sounds like Psoriasis... So really, no worries
You are washing your hair just fine, but this 'stiff' is more of a lump of dead skin cells. There is a treatment that can 'cure' it for several years (10, 20, or more) and it comes in like a small shampoo/conditioner container. You put it on the area 2-3 times a day and rub it in, then, when the area is normal, put it on one last time. The Psoriasis won't appear again unless major irratation occurs. I know a lot about this because my mom had this. lol
Anyway, just ask your doctor about it and suggest something for Psoriasis (pronounced SOR-RI-a-sis). Though my mom's stuff had cost around $200! (it really must work lol) so ask if med insurance can cover it!
Hope that does it