  We study the quench dynamics of the one-dimensional Hubbard model through the
Quench Action formalism. We introduce a class of integrable initial states --
expressed as product states over two sites -- for which we can provide an exact
characterisation of the late-time regime. This is achieved by finding a
closed-form expression for the overlaps between our states and the Bethe ansatz
eigenstates, which we check explicitly in the limits of low densities and
infinite repulsion. Our solution gives access to the stationary values attained
by local observables (we show the explicit example of the density of doubly
occupied sites) and the asymptotic entanglement dynamics directly in the
thermodynamic limit. Interestingly, we find that for intermediate interaction
strength R\'enyi entropies display a double-slope structure.
