Given the symptoms and the gamepad tester results plus what you've already tried, I do believe you do need to replace the potentiometers.
The problem cannot be caused by the joystick itself, as it does not have any connections. Only grounding pads for physical support. Replacing the potentiometers only could be a solution, but you would need the correct ones. Not all potentiometers are made equal and not all are compatible with the joysticks in your controller. It's generally recommended that you replace the entire joystick whenever possible, though not always required.