How the f**k are you supposed to drive these things? I'm trying to get past them so I can do the other oval events, but nothing I do can keep these damned monstrosities on the track. If I try and turn, I spin, if I don't turn, I spin, if I break, I spin, if I stay on the gas, I spin, if I ease up, I can almost turn, and then I spin. It's utterly maddening. Help!

I hear ya. I thought at first my wheel was all screwed up. But then it seemed hunky dory with the other cars.

The sprint cars pull left, and their back ends are heavier than the front, so they tend to want to spin to the left no matter what. As a result, you need to be turning right most of the time, even through the turns (counter-intuitive on a track that requires only left turns...) in order to keep the front end of the car in the front.
They don't give you a lot of margin for error though, so even having found out a bit about how they behave, I still find it ridiculously easy to spin out, hit 5 cars, both walls, and knock all four wheels off.

If you are using a keyboard, make light taps to the right so you can maintain momentum without crashing. I too thought it was a bad glitch. You can hit the wall and still continue the race as long as you don't lose your wheels.