Yes, kill the old grass if you want a Black Beauty lawn. If you can live with mostly old crummy grass, and 20 percent Black Beauty--fine power rake and have at it. You may get numerous spots of darker green Black Beauty. But you should also consider correcting the problem that caused the trouble--mainly drought and the lack of watering. If you are not going to water every day...you may result in almost zero Black beauty.
Fall is better for seeding (for cool season northern grass), wait until high temps come down below 85, (I would expect about Sept first for your area, NJ.)