I would use electrical
Contact cleaner.. Spray it on the Joystick it will work its way down the move the stock all around and then Blow it out with an air compressor.. Or compressed air..
Be carful what you use.. Make sure the cleaner is for small electrical components only! Iv seen some cleaners actually melt plastic..
If this worries you then watch a YouTube video of how the controller comes apart, take it apart and from there you can see the whole bottom of the joystick.. It's suppose to have a light coating of grease to help the joysticks glide nicely and smoothly, you can then clean it from there ( being able to get to all the sticky Soda from underneath) then take some clear dialectic grease and put it back on all the components of the joystick that make contact when moved.. This will help there be less friction thus giving you back a smooth and free joystick.
Sent from my iPhone using Tapatalk