Odd feeling just before I stop and just after I start

I have this odd shiver run through the truck while braking, but it only happens just before I come to a stop. Up until then the brakes feel normal and great, but that last 10', just before I stop, the brakes feel like they are pulsing. Not ABS pulsing but more grabby. And then, when I let my foot off the brake to move forward, the lightest tap on the gas feels like there is something slipping of shifting near the rear end. It is just a brief moment then everything is fine. Also when the engine is off and in park, when we rock the truck forward and backward it has a very odd feeling, much like the lug nuts are loose but they aren't. Nor are any other rear end bolts loose.

I know this is very tough to explain but has anyone had these issues and what you found the solution to be.

Thanks,
Paul