Your echeveria is not dying. As the plant grows it creates new leaves and the older ones die. Once they are completely dry, you can gently tug them off.
Also, make sure it's in fast draining soil. Echeverias seem to be one of the most sensitive plants when it comes to over watering. Even a couple days with too much water can start the rosettes to rot. When you water, soak it (soil only, not the plant) thoroughly a couple of times until it drains out of the bottom of the pot and then let it completely dry out before watering again.
They do enjoy full sun to avoid them from stretching out too much to get more light but appreciate being out of the summer afternoon intense sun.
