So, I too have an issue with one my joysticks, the right one specifically, where it will constantly push up, even thought it is centered. If I push slightly down on it, the stick stops moving, if I pull it down a little, it also stops moving. So I was searching for a fix for the problem, turns out Microsoft doesn't want really deal with it, and pretty much my only solution was to buy a new controller.
I didn't buy a new one, searching and searching for a fix, I took apart my controller and was playing around with the box the stick is situated in. If I take a small flathead screwdriver (used one from a standard eye glass repair kit), and stick it to the right of the joystick and push down, and exert a bit of pressure inwards (like you would do trying to pry out the stick), the "drift" so I decided to give it a little more pressure, heard a click, which sounded more like something slightly broken now, rather than something clicking into place. Pulled the screwdriver out, and bam, no drift, couldn't believe it. It will still "drift" upwards if I pull the stick right a bit, so not 100% fixed yet. It's not too annoying, because as soon as I push/pull it in another direction, the drift stops.
Anyone else want to try this and try and improve my results? I'm going to keep at it, but I think I'm on to something, or I'm going to break the controller. So much for MS testing this stuff.