Draven's early game is disgusting, but you have to keep in mind that as an ADC, you don't fight Draven. He fights you. Unless you are playing an ADC with lots of range and/or CC like Caitlyn, you simply aren't going to win trades with him 1v1. Instead, let him be your support's and jungler's problem. Your job is to bait him, not beat him.
In order to stack axes, he needs to repeatedly use them and catch them. This means that A) he is pushing the wave, making for easy ganks and free harass, and B) he is declaring where he's gonna stand to catch the axe. Zone him off from catching axes, and his damage output drops significantly, opening him up for punishment. Work together with your support to catch him when he's going for an axe.
Try to make skirmishes happen as early as possible. Draven is really bad at teamfights, especially early on, due to how high his W mana cost is. There's no shame roaming early instead of facing Draven head on. Force early fights over objectives or ping for help from your jungler, and you can overwhelm him.