I worked in a tire bay for a while. It sounds like your tires might not be balanced properly, that causes cars to shake a bit especially when at very low speeds. As for the car veering right, that sounds like an alignment. Check the wear on your tire treads to see if one side is balding faster than the other.
-hope this helps
could be a few things... could be tire related, air pressure, wear on the tires, etc. It could be alignment... have you hit any curbs or potholes? It could even be torque steer... though in normal driving conditions this wouldn't occur, though on a launch it is quite possible, usually doesn't happen in front wheel drive cars unless there is enormous amounts of power. I have this problem and I thought it was an alignment problem.
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