You'll definitely want to use Attack Move when playing an ADC, as accidentally walking a step forward when you meant to auto (even if you auto the wrong target) can be the difference in life or death. An ADC's life is all about positioning. I main Jungle and take ADC on the side as I enjoy someone more mechanical than Rek'Sai every now and again. Your job as an ADC is to stay alive and deal consistent damage to the largest threat you can reach. If you're only able to hit the Nautilus with 300 Armor because they have a completely long-range pick-oriented team and can easily collapse on you if 1 CC lands, that's what you have to do. It sucks sometimes in teamfights, but as you climb you'll notice your frontline tanks and supports will generally throw their life on the line to protect you, as you are the main consistent damage late game, as well as try to make engages that are favorable to your positioning. If I dive in 1v5 as Rek'Sai and their Rengar jumps my ADC, we will almost certainly lose that teamfight and I have failed our ADC as our damage is gone, while theirs is busy getting me peeled off of them and (more than likely) surviving the fight. Early game, most ADC's don't fair so well as their damage usually comes in the form of items whereas Tanks have %Health Damage to help shred you down that little bit faster. There are a few exceptions (Draven, for example, absolutely demolishes most champions at level 1 due to his sheer raw damage on Q), however most of the time you're desperately looking for gold to hit powerspikes quickly.
Your CS is incredibly important as an ADC hence why many tilt themselves into oblivion when they see a Lux support E'ing all the caster minions for herself. It's literally killing you and your team. You'll often find that your Support will dictate the early laning phase most of the time. If you're duoing with someone, it's a lot easier. In pick-up games of Solo-Queue, not so much. You'll have to learn a lot of cooldowns (i.e. Blitzcrank's Q is 20-16 seconds without CDR) and engage when it's favorable for you. If you have a Support that's incredibly aggressive early (say Morgana landing consistent Q -> W's on targets) you'll want to follow up (if you're able to) and force yourself into a dominating position to deny the enemy of valuable XP/Gold. If they're playing passive and safe, try to work around that and wait for your jungler to visit your lane. If they don't flex their power early game, you're going to have a bad time for the most part. This is often times while you'll see "SUPPORTAL KOMBAT" in LCS early game with both supports trading blows back and forth consistently. Both ADC's are watching health bars and waiting for that moment when they're able to sneak in and pick up the quick kill. A 2v1 rarely goes well afterwards, and often leads to a Double Kill and/or all summoner's burnt.
Since you like Draven, I'll try to give specific tips on supports. Draven enjoys bullying other ADC's around with his raw damage and dominating the lane from level 1. He will gladly trade blows with nearly any champion from level 1, as his damage is certainly higher with axes rolling. He's intensely mechanical as you almost require your axes to be viable, and your entire kit falls off if you drop them. With some practice with Attack Move, however, he's very similar to other ADC's and the axes can be very predictable. This works both for you and against you, as your opponent can see where they land and will often times attempt to CC you when you catch them - leading to a tough decision for you to make. Someone like Nami works overtime against Draven dropping Aqua Prisons (Q - Bubbles) on top of your axes when you go to catch them, and ruins your DPS and Mana from it. When in a teamfight and while chasing someone down your axes pretty much guarantee them a swift death, as it will continually reset your Blood Rush (W) giving you nigh unlimited movement speed. Draven synergizes incredibly well with a support like Thresh, who is capable of hard engaging very easily with Q -> E, and 1 hook often guarantees the enemy ADC's death with Draven backing up Thresh with his damage. Thresh also brings an escape (Lantern - W) for Draven, who is completely immobile, lacking any dashes or form of retreating safely aside from a short movement speed burst. This is often why you'll see Dravens run a QSS in their builds. Other supports to keep in mind are the likes of Blitzcrank and Leona, who provide a little less utility for Draven's safety, but still hard engage the same. Lulu and Janna are fine and will keep Draven safe and healthy, but to make the most of his Passive you'll want someone to hard engage for you or your jungler to camp your lane and get it snowballing. Draven does EXTREMELY well with a small lead - he's in a similar boat to Riven in that regard.
For Lucian, he's much more of a skirmishing duelist capable of kiting even the slipperiest of targets around. He synergizes well with nearly anyone, as he brings a lot of utility and self-peel in the kit of an ADC. He has resetting dashes between his Passive and E, and can fling out some pretty crazy numbers as well if you combo properly. His Ultimate is pretty great for clearing out waves or securing a kill from a moderate range, however it's very easy to dodge - with that being said if you're able to bait out all the enemy's mobility spells you can often times 100-0 squishier targets with it. Think of Lucian as a similar setup to Vayne, you'll often want to save your dashes for dodging the enemy team's spells rather than dive in and pray you get the kill. He's not necessarily as "safe" as an Ezreal, however he excels greatly in extended trades late game as he can continuously combo spells and put out nearly 50% more shots than your average ADC thanks to his double shot passive.
Try to practice your attack move in the Practice Tool against the Target Dummies. Put some in a bush and practice Warding to gain vision of them and hitting them as you move. This will save you countless times from the enemy ADC (and other champions) that turns on you and unloads at 10 HP when you just needed that one last auto and lost vision of them. Put a few dummies around you and practice running around them while still outputting damage. With the DPS shown, you can often land perfect combos while standing still, and then compare that to your DPS while moving. If you notice a marked drop, that's something you can definitely work on. If you're going from 1000 to 950 DPS, that's probably fine and completely normal, as we aren't perfect and can only time things so well. If you're dropping from 1000 to 500 DPS, that's a major issue in orbwalking/attacking and moving.
Hopefully this helps! Best of luck to you and I look forward to seeing you climb as an ADC! Feel free to ask any other questions if you have them, I'll do my best to answer them if I have the ability!
