Does it feel like the button is actually clicking when you press it? Does it feel correct? It sounds to me like either it's out of alignment or there some debris preventing it from pressing correctly. Definitely sounds like you're in from a good tear down and cleaning. It can be a lot of work, but trust me it's worth it. Much better than a whole new controller.
There isn't a full teardown here at iFixit, but you can get pretty close just using the [guide|144120|bumper switches guide|stepid=291921|new_window=true].
There is a little membrane that makes contact with a spot on the board and a tear in this membrane or just some crud between the contacts can interfere with the button function.
This one specifically is the A button, but cleaning this part of the board up with some isopropyl and a cotton swab might get you back in business. Check the membrane too.
[image|3473412]