Just bought a PS2 slim console so I could still play my PS1 games (Chrono Cross & FFIX, specifically) and open up new possibilities with PS2 RPGs. Got everything connected, fired up my Chrono Cross (PS1) game, things were looking good. You have to press the analog button in order to use the dualshock stick with this game, it doesn't automatically come on when you start the game. When I did this on the PS2 controller, Serge started walking off all by himself. Turn analog off and everything's fine using the D pad, but I need to be able to use the analog stick (arthritis in my hands). Plugged in FFX and it even started to do it with that game too (Titus started walking off all by himself), but stopped and seemed to work fine after that. Plugged in the controller for my PS1, and Chrono Cross worked fine. Should I return the PS2 controller?

Yep, sounds like drift. You can try cleaning it, in case there's dirt/debris causing it, or swap it for another one.

Thanks Dan, I appreciate your input. Thanks also for attaching an actual name to what this controller is doing. I've been playing games for several years now, this is the first time I've experienced drift in the controller.