If you are careful you can leave the two sided tape under the grips intact. Poke holes through the two sided tape to get to the screws near the outside edges. Getting the grips back on can be a chore to press the nubs back in the holes. Just keep working them, they will go in. Once you have the back of the ST16 off make note of the electrical plugs to the gimbal and which way they go back. I found it easier to disconnect the two connectors to the gimbal, unscrew the 4 screws on the front and completely remove the gimbal, then I could thoroughly spray the pots with plastic safe electronic component cleaner over a garbage can, then move the stick rapidly through all positions and repeat.
Once you reconnect it screw the gimbal back in place from the front then test the ST16 to see if the outputs are giving you full deflection on the bad stick, if not repeat the cleaning. Once you know its clean and working, then completely replace the screws and grips.