I play Graves often and I also find myself going up against Ezreal a lot. The lane is not as heavily favored towards Graves as you feel it is, and really supports play a big role in the matchup.
There are 2 pieces of advice already stated in the replies that you seriously need to take.
First off you can blink dodge Buckshot using your E to get out of the way. Since Buckshot comes out in a cone/triangle shape, that means the most effective way to blink to maximize your chances of not getting hit is TO THE SIDE AND TOWARDS HIM. Keep practicing this and don't hesitate, reaction speed is very important in dodging skillshots.
Second thing you need to do is stop building Black Cleaver on Ezreal. If you build Bloodthirster you will leech life back and you will last longer in fights, therefore increasing your chances of winning the fights. There are a few different builds that work effectively on Ez, but none of them really involve Black Cleaver. I'm not necessarily saying BT rush is the best (lots of people prefer Triforce), I am just suggesting you to upgrade the BF Sword you get into a Bloodthirster, which is a much better choice than a Cleaver.