I had a thought. My first thought was carb floats, and it still may be something wrong with them. But after you said braking, esp when you brake hard, I think you may have a vacuum leak in your braking system, that is, if you have power brakes. I think it maybe a problem with both the brakes and the floats. Try different things with the brakes, such as breaking with the vacuum line pinched off and see if the engine still dies.