Contact cleaner. DeOXit is the one I use most often.
You dont need to replace or solder anything. Take the controller down to the point that you can take the sticks off. Pull them off. Blow a quick spritz of deoxit into the middle of each assembly, waggle the stick around a bit, wipe/blow off the excess, and there you go.
I have no idea why everyone seemingly forgot this around 2015 or so. Analog controllers have been the same arrangement of X/Y potentiometers since like 1950 something, they have always gotten dirty, which manifests as "drift", and contact cleaner has always fixed it.