A car will be bouncy because the shocks are bad. Springs that are too hard beat up the car, as well as cause the
car to lose grip with the road on the bumpiness of the street. If you personally are cutting out so much of the spring that you are significantly changing the spring rate, maybe you should go back and do some measuring of the suspension. If your car is resting on the bottomed-out shock instead of the springs, an aftermarket spring of the same height would also cause your car to rest on the bottomed-out shock.
Your next problem is that taking a spring & removing some of the coils will cause the spring to become much stiffer (less wire = harder spring). This is why your car will be Hella bouncy. The severe stiffness of the springs can't be overcome by the shocks & will cause them to be destroyed. Now you need new shocks, and that cost much more than really good lowering springs.