  The compounds BaDy$_2$O$_4$ and BaHo$_2$O$_4$ are part of a family of
frustrated systems exhibiting interesting properties, including spin
liquid-type ground states, magnetic field-induced phases, and the coexistence
of short- and long-range magnetic orders, with dominant one-dimensional
correlations, which can be described as Ising $J_1-J_2$ zigzag chains along the
$c$-axis. We have investigated polycrystalline samples of BaDy$_2$O$_4$ and
BaHo$_2$O$_4$ with both neutron diffraction and neutron spectroscopy, coupled
to detailed crystalline electric field calculations. The latter points to
site-dependent anisotropic magnetism in both materials, which is corroborated
by the magnetic structures we determined. The two systems show the coexistence
of two different long-range orders --- two double N\'eel
$\uparrow\uparrow\downarrow\downarrow$ orders in the $ab$-plane with
propagation vectors $\mathbf{k}_1$ = ($\frac{1}{2}$,0,$\frac{1}{2}$) and
$\mathbf{k}_2$ = ($\frac{1}{2}$,$\frac{1}{2}$,$\frac{1}{2}$) for BaDy$_2$O$_4$,
and two distinct arrangements of simple N\'eel
$\uparrow\downarrow\uparrow\downarrow$ orders along the $c$-axis, both with the
propagation vector $\mathbf{k}_0$ = (0,0,0) for BaHo$_2$O$_4$. The order for
both wave vectors in BaDy$_2$O$_4$ occurs at $T_\mathrm{N}$ = 0.48 K, while in
BaHo$_2$O$_4$, the first order sets in at $T_\mathrm{N}\sim$ 1.3 K and the
second one has a lower ordering temperature of 0.84 K. Both compounds show
extensive diffuse scattering which we successfully modeled with a
one-dimensional axial next-nearest neighbor Ising (ANNNI) model. In both
materials, strong diffusive scattering persists to temperatures well below
where the magnetic order is fully saturated. The ANNNI model fits indicate the
presence of sites which do not order with moments in the $ab$-plane.
