Stick drift is the controller problem of this console generation, and it has three different causes that look identical from the couch.
First, rule out calibration. Most systems have a stick test in their settings or controller menu. Open it and let the sticks sit untouched. If the dot rests off center but the movement is smooth, try the system's recalibration. Some drift is just a baseline that drifted with wear.
Second, debris. Grit around the stick base causes jumps and dead spots. Isopropyl on a cotton swab at the gap, work the stick in full circles, and let it dry. It is worth fifteen minutes before any soldering.
Third, the potentiometer itself. The stick mechanism is two carbon-track pots, and they wear. Contact cleaner can buy months. The permanent fixes are replacing the stick module or moving to a hall-effect stick, which removes the wear surface entirely. Both are soldering work, and both are real repairs people do at home with practice.
A note on honesty: if the controller is twenty dollars and the repair needs a soldering iron and an hour, the math is yours to make. Drift on a favorite limited-edition pad is a different decision than drift on a pack-in controller.
What fixed yours, or what did you try that did not? Include the model; drift rates vary wildly by revision.