Chosen Solution
I have had this controller for a few years and it worked fine for at least one of those years but now the analog stick sunk inward and it doesn’t work like it used to.
Try taking it apart and fixing it. This is Ifixit you know. Update (12/31/2017) Here is the link for the repair guide: DualShock 3 Analog Stick Covers Replacement
Have you tried pulling it out again? (This may sound ridiculous but worth a shot)
Just did this.. Pulled thumb cap off stuffed some aluminum foil in the hole and put back on