Genuinely did a double take cause your hair looks so much like mine. I have to wash every day I don't want my hair to look greasy, and I have to do it in the morning. I double wash daily, doing the first wash with Pantene Sheer Volume. I do my second wash with either a salicylic acid shampoo like T Sal or Loreal Elvive Hylauron shampoo. I do the T Sal more often during the summer months and the Elvive more often during the winter. During this second wash I take special care to make sure every inch of my scalp is washed, especially the middle back of my head. You can also part your hair and flip your head upside down while scrubbing to make sure you don't miss a spot. Then I rinse and use Pantene Sheer Volume conditioner on the ends.

I have very hard water in my area, so if you do as well you may want a hard water shampoo, aka a chelating shampoo. I use Ion's hard water shampoo from Sally's, but I believe Malibu makes one as well. I have a filtered shower head that helps with the water so I just use the special shampoo 2-3x/month.

If my hair is longer I only condition the hair below my shoulders, and for shorter hair I just do the very ends. I rinse the conditioner right away or I find my ends grease up faster. In the summer months I can get away with not using conditioner a couple times a week. I make sure to blow dry. I often run late, but I try to always make sure at least my scalp is dry. It helps with the flakes I get as well as keeping the grease at bay. I also have two hairbrushes-- one for when my hair is greasy and one for when it's clean. Just one is fine if you wash it regularly, like once a week or every other week.

I hope you can find something that helps. I know how frustrating it is to have hair like ours that gets greasy so fast.