I had trouble configuring NFS:Underground and NFS:Most Wanted exactly the way I wanted them with my preferred analog joystick, (had to reverse-engineer and hand-edit the binary config file for Underground, in fact), but it was worth the trouble. Those were two of the most fun games I've ever played.

Our difficulty getting certain controls to work with those particular games is not unusual, from what I understand. I know know why the controller input configuration on the NFS games is significantly worse than other games, but it is. Ah well.

Instead of a wheel controller or a keyboard, try a plain old analog gaming joystick. One of the ones with thumb controls, like the ones for playstations and xboxes and such. That's the way those games were meant to be played anyway.

I've been using a Nintendo Gamecube controller plugged into a USB port adapter, and aside from the configuration issues I've described, it's been really great for arcade-y games like that. I assume other joysticks of that general type would work similarly.

Regarding the "always drift to the right" keyboard steering, is it because mouse steering is turned on and your mouse is drifting? In any case, don't bother trying to play with keyboard steering, that's just no fun.