I would like to comment based on my experiences... I'm definitely not a nutritionist!!
The type of fat can play a big part in dry skin. Omega 3/6/9 are the best. Polyunsaturated fat is the worst. I used to eat a lot of fat and when I started taking Omega 3/6 supplements my skin and hair got a lot better.
Tee tree oil can make it worse. It does not moisturize at all. I used to use it on my face and it wreaked havoc on my skin after an initial improvement. Your skin needs natural oils to stay healthy.
You could also have another issue causing the dandruff, have you seen your doctor? My son has cradle cap (at 4 years old), I think it is a type of fungus or infection - but someone else might be able to tell you exactly what it is, and we massage olive oil onto his head before washing his hair every few months and it seems to keep it at bay. His dandruff gets really bad if we don't do this.
How often do you wash your hair? I would suggest that hair only needs to be washed every 2 or 3 days so that the oils can keep everything moisturized.
I hope this gives you a few ideas. It may not be your diet at all, but it's a good place to start!