When I was a little girl, my grandmother would wash my hair with shampoo, rinse it and then pour a solution of one cup of 1/2 vinegar and 1/2 water on my hair and then made the final rinse with cold water.
This solution left my scalp healthy and hair beautiful. The underling cause of dandruff can be a fungus. Vinegar kills fungus.
Also, it is helpful to use the pads of your fingers to scrub your head. Washing a person's hair is not so much cleaning the hair as cleaning the scalp and rinsing properly.
If this doesn't clear your problem up in a few weeks, I have found that my head feels better and doesn't itch or flake with Garnier Fructis Fortifying Shampoo, Sleek and Shine. It has vitamins B3 and B6, Apricot and Avacado Oil. Just make sure that you rinse, rinse, rinse, until your hair is squeky clean.
I LOVE this shampoo. I have no idea if it is vegan, though.
Also, hot water can dry out the scalp, so its better to wash with cooler water.
Hope this information was helpful.