I'm guessing your rear u-bolts are loose, and causing the rear end to shift(axle wrap) and bind on the rear drive shaft. Because you are not directing as much torque and h/p when you are taking off slowly, it doesn't shudder as much. But when you stomp on it, it places lots of strain and movement on your rear axle and places strain on the drive shaft. The shuddering while turning is due to the driveshaft letting up in the other direction, when the axel wrap occurs while slowing down.
Or maybe Pea-Pie is stuck in the grooves of your tire causing it to be unbalanced!
Ric-H0