The Fomin local rules for Schensted column insertion can be seen as a two-lane box-ball system, in which a carrier moves particles forward or laterally. Running such two-lane dynamics in parallel on a periodic lattice gives rise to a two-dimensional generalization of the box-ball system, which we call the \emph{skew column RSK dynamics}. Equivalently, this is a deterministic dynamics on pairs of skew semistandard Young tableaux $(P_t,Q_t)_{t \in \mathbb{Z}}$. We prove that this dynamics exhibits solitonic behavior and construct an explicit bijection $(P,Q) \leftrightarrow (H_1,H_2,κ,ν)$ that linearizes the time evolution. The resulting coordinates consist of two horizontally weak tableaux $H_1,H_2$ recording the asymptotic soliton data, integer riggings $κ$, and a weakly decreasing sequence of integers $ν$. A key feature of the construction is an explicit projection from the skew column RSK dynamics to the classical box-ball system; under this projection, the rigging $κ$ is precisely the Kerov--Kirillov--Reshetikhin rigging of the associated box-ball configuration. Our proof uses two commuting affine crystal structures on pairs of skew tableaux and a novel connectivity theorem for distinguished subgraphs of tensor products of Kirillov--Reshetikhin crystals. We also derive Greene-type formulas for the soliton lengths in terms of last-passage percolation on the associated cylindrical environment. Finally, by taking generating functions in the linearizing coordinates, we obtain bijective proofs of Cauchy and Kawanaka--Littlewood-type identities for transformed Hall--Littlewood polynomials.