  We compute almost-complex invariants $h^{p,0}_{\overline\partial}$,
$h^{p,0}_{\text{Dol}}$ and almost-Hermitian invariants $h^{p,0}_{\bar\delta}$
on families of almost-K\"ahler and almost-Hermitian $6$-dimensional
solvmanifolds. Finally, as a consequence of almost-K\"ahler identities we
provide an obstruction to the existence of a symplectic structure on a given
compact almost-complex manifold. Notice that, when $(X,J,g,\omega)$ is a
compact almost Hermitian manifold of real dimension greater than four, not much
is known concerning the numbers $h^{p,q}_{\overline\partial}$.
