A coded shift is the closure of all bi-infinite concatenations of words from a fixed countable generating set. It admits a natural decomposition into a concatenation set of genuine concatenations and a residual set arising from the closure. We prove uniqueness of equilibrium states for Hölder potentials when the pressure of the concatenation set dominates that of the residual set, and, under mild growth assumptions on the generating set, show that the resulting equilibrium state exhibits exponential decay of correlations, satisfies the central limit theorem, and obeys the law of the iterated logarithm. Notably, these statistical properties hold under considerably weaker hypotheses than uniqueness itself, and remain valid even when the equilibrium state is not unique. As an application, we establish all three properties for the (possibly non-unique) equilibrium states of Motzkin–Dyck shifts — new results even for measures of maximal entropy.