Update Again...
Got the two 'for parts' controllers in today. Just for a lark, I tested them out before tearing them apart. One was recognized by the system as being plugged in, but none of the buttons worked at all. Surprisingly, the other one works perfectly, so that was a nice little bonus.
Took apart the non-working on to harvest the foam piece from and get my other controller working again. Took said foam piece and put three layers of duct tape on the bottom, then reassembled everything. It works flawlessly again. So, anyone who's have issues with random or spotty button presses in a Dualshock 3, give that a shot before junking the controller. Just layer some thick tape on the bottom of the foam, cut to size, and put it back together.