I carried myself from 1200 - 1420 back to 1200 (after a week of not playing) and back up to 1460, all by playing Anivia (whom I have a 70% win ratio with). My ingame username is just 'Otterpaw' if you want to look me up.
As you said, you should play carries. You CAN carry games as tanks and support, but its far more difficult, as you have to hope for your teammates to take advantage of the opportunities that you present.
Therefore, as a carry, you have to make several assumptions about your team.
1st: They have no damage-
If you are the only person last hitting in the match, its more than likely that you are going to be far more farmed than they are. If your team is worse, the rest of the team is far more behind in gold. DONT rely on their damage, instead
2nd Treat your teammates as nothing more than sacks of meat between you and you're enemy. They're handy as a wall, but until you get up in elo a bit, they're not going to be half as strong in the match as you are.
3rd Your allies are idiots. They're going to overextend. They're going to make retarded tower dives. DONT GET ANGRY. DONT SHRED YOUR TEAM'S CONFIDENCE AND MORALE. A team determined to win has a far stronger chance than one that's arguing within itself.
4th They won't ward. Do it yourself. You should be the carry. You should be able to afford wards with your farm.
5th Control dragon and baron. Your team won't farm. Their only source of gold will be kills, dragon, and baron. You need to take control of these map objectives (with wards) and coordinate your team to take them by
6th Pinging like a madman. That's the only way to get your teammates to listen. Ping pingping ping. or scream in chat (caps lock IS your friend). Again, don't chastize when they fail to do so.
Things you have to do:
1- FARM FARM FARM -- if you don't know how to last hit, learn to! Pushing your lane is a BAD BAD BAD thing.
2- Command the noobs. Control their every movement. Don't retroactively tell their stupid move was a bad decision. Tell them the good decision BEFOREHAND, and why the wrong decision was wrong. If you don't tell them what to do, they're going to be stupid.
3- Carry the team. This should be obvious. Stand behind the meatbags, and use them for cover as you rip through the enemy team.
4- DONT DIE - if you die, the only source of damage your team has is gone. I was carrying a game 14-3-7 as anivia. It was lategame, I facechecked a brush (stupid move on my part) and died. 4v5 a team SHOULD be able to hold a tower, but as I've said, I was the ONLY damage on the team. They got 2 towers and the inhibitor (due to an incompetant team), and it resulted in a lost game.
5- realize that YOU MAKE MISTAKES. You are not a flawless player that should, in fact, be in the 2000's. You are a scrub, just as I am. We all make many mistakes, and if you see yourself make a mistake, take the blame. If you constantly blame others for your mistakes, you will NEVER get better. I made a terrible mistake in #4, when it could easily have been avoided by tossing a wall in the brush and playing more conservatively. I made that mistake, I took the blame, and I learned from it.