The evolution of a supernova remnant through the transition from an adiabatic Sedov-Taylor blast wave to a radiative pressure-driven snowplow phase is studied using one- and two-dimensional hydrodynamic simulations. This transition is marked by a catastrophic collapse of the postshock gas, forming a thin, dense shell behind the forward shock. After the transition, the shock front is characterized by a deceleration parameter, Vt/R ~ 0.33, which is considerably higher than the analytic estimate of 2/7 for a pressure-driven snowplow. In two dimensions, the catastrophic collapse is accompanied by violent dynamical instabilities of the thin, cool shell. The violence of the collapse and the subsequent instability of the shell increase with increasing ambient density. Preshock density perturbations as small as 1% in an ambient medium with density of 100 cm-3 can lead to distortions of the shock front larger than 10% of the radius of the remnant.