Unfortunately, both your zoom lenses are quite slow. By contrast, a mate of mine uses a 400 f/4 or 200 f/2 for his action photography. You are right about the prime lens. It would perform much better, but of course you would need to crop later. If you're up-close however, it may be an option.
With your current setup, I'd say to use the speed priority as you mentioned. Depending how close you are to the action and relative speed of the cars, use a shutter speed of around 1/320th if they are quite far, or 1/800th-1/1600th if you are close up. Use auto-ISO and let the camera choose the aperture and ISO.
In open conditions like this, a flash will be no good to you whatsoever, as the flash range just won't be enough. This will be compounded by the camera being 'tricked' into thinking whatever it is taking a photo of will be lit up by the flash and will result in even more under-exposure.
Make sure you shoot in RAW (this is the only reason I don't suggest the built in 'sports' mode - I think that will only capture in JPG?). RAW means that in post, you'll have better chances of suppressing any noise and balancing grain with sharpening.
Also make sure you use continuous servo AF. I think the D3100 also has some kind of 'predictive' AF - so read up in your manual on using that. But the continuous AF will ensure it keeps the car in focus as you're panning, so helping get a better photo.
Finally, use spot metering so your camera knows to expose only for the subject under your AF point. This will help ensure you get the right exposure for the car - not the overall gloominess of the skies!