I had similar problems with Happ Comps, but found that there were many people on this forum couldn't understand the problem - However, those people usually liked to play competitive fighting games, where the problems are not really apparent.
Unfortunately, I found that I couldn't play many favourites like Ghosts n Goblins because of the stick "twanging" back and hitting the opposite direction. In GnG, Arthur would often uncontrollably spin around 180 degrees after a volley of spears into a baddie, which pretty much got him killed a lot.
I solved the sensitivity problem using a large metal washer on top of neoprene rings cut from neoprene pipe insulation. This all went underneath the joystick's spring to tighten it up slightly, and made it act more like a joystick with a rubber centreing grommet. For details, look at this thread:
http://forum.arcadecontrols.com/index.php?topic=74087.0Basically, I put the shims into the P1 joystick only, as most of the games affected were single-player games. With the shims, Arthur was very happily in control of himself and got to kills many ghosts, zombies & demons. I found no (or neglible) impact on playability of fighting games, so I call this a success.
The cab in question was done for a customer - he has been very happy with it since.
Cheers, Zeb.