This sounds a lot like me as a teen. I had very thick, coarse, dark hair that always looked greasy, along with a helping of dandruff. What I've found personally is that I needed to wash my hair less often. It sounds counter-intuitive but I wash my hair only every second day now and as a result, it becomes oily much slower.
Other things that have helped me: Neutrogena T-Gel as a previous poster suggested. I use this a couple times a week.
In order to get the shampoo entirely through my hair, sometimes I shampoo, rinse, shampoo rinse, before I condition.
I found areas of my hair were oilier than others, specifically the back of my head. I wasn't getting the shampoo all the way into that area. I spent more time applying shampoo there, and had to learn to really reach my scalp in all places.
Finally, I asked my hairdresser for advice. He was really helpful in helping me learn how to tell a) when my hair was thoroughly covered in shampoo and b) when it was completely rinsed.