I'm an ADC main that can play Draven well enough to carry if that's close enough?
Laning phase: A little background context, this used to be hella broken back when he had his damage passive. When they replaced it with a gold generating passive, people actually had to think about what they did. Sad day, I know. So Draven has two major power spikes. Level 2, and again when you get BF sword. Level two is because you can take E second; Q-> E->Q and the E prevents them from meaningfully retaliating. If you're soaking up minion aggro in excess then that means one of two things. 1) You're going too ham to harass or 2) you're trading for too long (closer to skirmishing). You rarely want to sacrifice farm for harass. Stay within your minion wave and auto them in between your csing when they're trying to get theirs. Double axes Draven always wins trades, short trades to be exact. If you get two Qs off, and they get two autos off. You're going to come out on top in terms of damage. Don't stay for the rest of their spell rotations. Drop their hp bars, back off to farming, and repeat until they're either forced out of lane or you can kill them (with your support). Don't ever try to initiate an all-in. If you have a support like Leona, let her lead and make use of her passive procs for dmg. Once you get your BF sword, you can look to all-in but play it smart and don't just force it through a bad engage.
Post laning: This is pretty much standard ADC stuff. Stay in the back lines and don't overextend. Just kill people as they come to you. If you can get a high priority target without putting yourself in imminent danger, go for it. Otherwise, it doesn't matter who you're hitting as long as your outputting dmg from relative safety (You're going to do large amounts of damage, even to tanks). Look to pick up double axes before team fights start, whether off a minion wave, nearby jungle creep, scuttle crab, etc. You can sustain it even longer if you just refreshed it with a catch and have the Q off cd (because then you can activate Q when its almost gone to refresh it). I play greed axes and build mostly AD, but the norm is typical AD stuff. IE->PD->BT->LW, etc etc. Build according to what you need. If you can catch your axes in team fights, sweet. If not, just use regular autos until one comes back up.
General tips:
- Do NOT overextend just to catch an axe. It will never be worth it, and will get you killed or massively out-traded most of the time.
- You can, to an extent, control where your axes land by the direction you move between the time you threw it and the time it hit them. So if you throw it and move backward, the axes will land behind you.
- The ideal time to W is right after you catch your axe, so that you can catch up to the target. For the reason, you want to start a little closer to your target than a normal ADC (the movespeed boost can only cover so much)
- If you have your E up when you get ganked and the jungler is coming from behind, use it on him. If he's coming from the side, use it on your enemy laners. 2 out of 3 is disrupted is ideal, otherwise the highest priority target.
- As said above, don't force extended trades unless you know you have the damage and mechanics to destroy them. Just force them to eat your axes in short trades until they're too low to do anything and can get zoned.
- Your ult is an execute, use it as such. If you start the fight with your ult, and they end up surviving/running. Guess what you're going to snipe them with? Nothing because you already ulted for cheese damage when you could've used your Q damage to win. The only exception to this is a perfectly lined up enemy team at the beginning of the team fight (even then its pretty controversial), or if you're not there when a team fight breaks out and need an immediate contribution to give until you get there.
- To be honest, don't be afraid to buy wards yourself and use them. It makes a huge difference in seeing ganks coming ahead of time when placed in the right spots, and it's next to no cost to someone like you with a consistent gold income from farming. Definitely upgrade your trinket as soon as you can for the mini, recharging sightstone.
In terms of your support: If they don't want to go in, guess what? Don't go in. You don't need your support for short trading, which is how most lanes are won, and what Draven excels at. last hit, pop them with an axe when they go in to last hit, go back to last hitting yourself. It will add up, trust me. When your support does finally decide to go in, your pattern is the same. Q->E->W to catch up->Q->W->Q until the skirmish is over or until they're too far away to continue.
Hope this helps.
EDIT: I can probably snag one of my friend's EUNE accounts for a couple games, or just make one myself and use whatever's on free week.