Skidding will not make you stop faster, in fact, it will increase your stopping distance. I've locked up the rear wheel under hard braking a couple times trying to avoid idiot cagers but quickly let up on the pressure to regain control.

As for your fit problems, is it just your hands or are your shoulders/back also sore? If it's all of the above you might try a shorter stem and moving your seat back a bit. If it's just your hands I'd suggest gloves if you don't currently wear them. If you are too close to the handlebars you will put more weight on your arms/shoulders with the seat farther forward. Try moving it back a bit and see if your comfort improves.

Road bikes take a bit of getting used to. Riding in the drops will not be comfortable right away, you have to get some mileage on the road bike to get your body used to the position.

I have two road bikes and rarely do I ride in the drops. I've ridden a century on one of them with no soreness afterward.