Getting your weight behind the seat is a means of unweighting the front wheel and restoring the neutral balance of a flat slope. You should feel about the same pressure on your hands as when you are just riding along, and if you are REALLY shifting your weight, you might even feel like you are pulling up on the bars.
"Leaning back" is probably not a good way to describe it. "Shifting weight" is much more accurate. If you "lean back" you might instinctively start with your shoulders, which would not shift much weight, and would require enormous leverage to move the front of the bike. If you use weight closer to your center of mass (your giant ass, for example ), you will shift weight more quickly with less effort.
Instead of thinking about your hands, I would try thinking about your feet. Do you use clipless pedals or toe-clips/straps, or platforms with some angry teeth? I ask because I wonder if you have confidence that your feet are in control on the pedals. Are you pressuring your feet to stablize the bike on descents?