I want to second the use of vinegar -- cider or white will do. We wash the hair, towel dry it (no rubbing) and then I work the vinegar into the hair; be sure to get all the way down to the scalp. We leave it on for a while as my daughter reads etc. and then rinse it out very thoroughly. I prefer to do it on a weekend so the vinegar smell doesn't linger while she's at school (someone else posted that the smell is gone once the hair is dry but that's not the case with my daughter, who has very thick hair).
Vinegar helps get out oil, but for dry scalp you may need to do the opposite: Warm olive oil in the microwave (not too hot) and apply it to the scalp while using a fine comb to rub it in and simultaneously gently scrub up flakes. Then follow with a good wash and the vinegar treatment. Takes time but is a natural way to deal with the flakes, which may be more from dry scalp than true dandruff.
And yeah, unfortunately one can have both a dry scalp AND oily hair.
Good luck. My daughter is 13 and I'm always seeking a good, simple shampoo to help with the oily hair day to day but not dry the hair and scalp out. Neutrogena Clean shampoo is pretty good. I have used T-Gel myself many times but I would not use it on a kid -- it is strong stuff, and if it gets in her eyes or mouth even once she is going to run a mile before using it again. Also it's coal tar shampoo, which you really don't want on skin any longer than absolutely necessary. I'd go with the most natural things you can, though they require more work (like the vinegar and oil treatments) and a very basic shampoo with minimal chemicals. T-Gel and Head and Shoulders et al are really for serious dandruff and she may or may not have that.