  We consider the totally asymmetric simple exclusion process in a critical
scaling parametrized by $a\geq0$, which creates a shock in the particle density
of order $aT^{-1/3},$ $T$ the observation time. When starting from step initial
data, we provide bounds on the limiting law which in particular imply that in
the double limit $\lim_{a \to \infty}\lim_{T \to \infty}$ one recovers the
product limit law and the degeneration of the correlation length observed at
shocks of order $1$. This result is shown to apply to a general last-passage
percolation model. We also obtain bounds on the two-point functions of several
$\mathrm{Airy}$ processes.
