  The study of quantum quenches in integrable systems has significantly
advanced with the introduction of the Quench Action method, a versatile
analytical approach to non-equilibrium dynamics. However, its application is
limited to those cases where the overlaps between the initial state and the
eigenstates of the Hamiltonian governing the time evolution are known exactly.
Conversely, in this work we consider physically interesting initial states for
which such overlaps are still unknown. In particular, we focus on different
classes of product states in spin-$1/2$ and spin-$1$ integrable chains, such as
tilted ferromagnets and antiferromagnets. We get around the missing overlaps by
following a recent approach based on the knowledge of complete sets of
quasi-local charges. This allows us to provide a closed-form analytical
characterization of the effective stationary state reached at long times after
the quench, through the Bethe ansatz distributions of particles and holes. We
compute the asymptotic value of local correlations and check our predictions
against numerical data.
