This is often caused by dirt or wear in the analog potentiometers. A quick blast of canned air or electronic contact cleaner (like WD-40 Specialist ) can sometimes clear it up without soldering.
Use a Q-tip dipped in 90% isopropyl alcohol (IPA) to clean around the button housing. Press the button repeatedly to let the liquid break down grime. xbox 360 controller schematic pdf hot
details how to strip and resolder these connections for USB conversion. Analog Stick Circuitry This is often caused by dirt or wear