I'm a Draven otp and I'm in no means considering myself good (hardstuck g1/2), and I don't think draven "falls off" because he still is an adc. As long as you keep up in cs, don't feed, and if your team has some ap you should still be putting out damage, if you have a mountain drake, look to get pinks and sneak a baron or obj etc you'll one shot it. Draven can still 3-4 shot carries as long as you have the correct build , and as long as you position well you'll still dish out decent damage. In fact if you're behind it might be good for you because the enemy will probably focus somebody else. Even a late game vayne vs a full build draven will always lose a 1v1 unless shes got rb stacked and condemns you against wall. Never hesitate to go tabi on draven vs talon/khazix/yi. It will save you a lot, and the more you live the more damage you dish out.
I personally perma ban cait, so I can snowball lane. Typically always look for fb on the level 2 powerspike. Then enemy bot never expects it in this elo, however sometimes your support might not understand it so just tell them before hand, if they stay and your mechanics are average you will at the very least get one or two flashes, and a heal. Don't bomb your flash too unless its a for sure kill. Now you have their flashes down and look to tell your jungle, "freezing bot no summs free kill", and until then just harass. Draven always wins trades as long as your don't eat abilities or you're against a cait.
One thing harder to do on draven when you first start playing him is map awareness, make sure if you're shoving, that you ward and look at map every couple seconds. If you're ahead and comfortable with 2v3ing with the enemy jungler, shove. However if they have lets say a talon mid or a fed jungler, or even a non fed jungler that you're just not comfortable with, don't push because draven mains say you have to push. Better to freeze and win game rather than tilt and feed.
I recommend to watch jackspectra's league basics guides. He will tell you all about match ups and how to play alongside certain supports and how to properly cs trade, control wave, teamfight, etc.
Lastly I was told this by a Diamond player, learn to dodge and multi query op gg before games. I had a couple troll supports in champ select today and I'd rather lose 3 lp no mmr or even 10 lp no mmr than 18 lp and mmr. Dodging is an important skill as well, it takes patience but it's worth it. Even if you don't have a troll support but your comp doesn't match up to well, it's always safer to dodge.
Also don't tilt or flame, turn your chat off if you can't help yourself and turning it back on once you have a lead and want to make objective calls. People will listen to you that way, especially if you're carrying and don't say "Do baron monkeys". Nobody listens to a toxic player, if you're not muted by that point.