Baffling. Your CPU isn't overheating, is it? Maybe even the GPU? Are all fans spinning?
It really sounds like some sort of bottleneck in system traffic, which is why I thought it might be an AGP problem. I dunno, try Display Properties (right-click desktop > Properties) > Settings > Advanced > Troubleshoot and try alternating running a 3DM01SE or dxdiag test and changing settings, one at a time. I'd try disabling write combining first, then lowering Hardware Acceleration a few notches (one at a time), finally disabling Fast Writes (via SmartGART). I left FastWrites for last b/c IIRC (a search here may turn up something) the 9600 series either required them or had no problem with them, unlike some other cards of their time. You could try disabling FWs first, just to get them out of the way.
The fact that even the DirectDraw test stutters may be a clue for someone more knowledgable about these sorts of system quirks than I. Four notches down the Troubleshoot slider disables hardware (GPU) DirectDraw and D3D acceleration, so if you still see stuttering then I believe it's _not_ a GPU hardware/driver problem. I still think it's a problem with either the CPU or CPU-GPU communication, not necessarily the GPU itself.