A recent study [Mora et al., 2026] has shown that the free fall of a sphere through the drag crisis can be modeled as a two-state stochastic process, leading to intrinsic velocity unpredictability. This dynamics is governed by a system of two coupled Kolmogorov partial differential equations for the probability densities Pα(u, t) and Pβ(u, t) of the velocity according to the two possible subcritical and supercritical drag states. While previous investigations were restricted to the weakcoupling limit where transition rates are small, the present work extends the analysis to the general regime. Through numerical simulations, we identify the emergence of singular probability density divergences at the terminal velocities uα and uβ. Exact analytical stationary solution demonstrates that the qualitative singular behavior is robust, regardless of the specific functional forms of the transition probabilities. The divergence is an asymptotic feature reached only as t →∞ due to the logarithmic relaxation time toward the attractors. These results elucidate how the interplay between the switching frequency and deterministic advection toward the terminal velocities dictates the asymptotic statistics of such bistable hydrodynamic systems.