  Cobaltates with 3$d$ based layered honeycomb structure were recently proposed
as Kitaev magnets and putative candidates to host the long-sought Kitaev spin
liquid. Here we present inelastic neutron scattering results down to 50 mK for
powder samples of Na$_3$Co$_2$SbO$_6$ and Na$_2$Co$_2$TeO$_6$, with high
resolution in regions of low momentum and energy transfers. We compare the
experimental data below the antiferromagnetic zigzag ordering temperature with
dynamical structure factors obtained within spin wave theory. We search the
wide parameter range of a $K$-$J_1$-$\Gamma$-$\Gamma^{\prime}$-$J_3$ spin 1/2
model and identify the best fits to constant momentum cuts of the inelastic
neutron data. The powder average limits the selection of a unique parameter set
for each material, but we see clear trends towards ferromagnetic Kitaev
exchange in both compounds, and ratios $|K/J_1|$ could be as large as $5\ldots
25$; in contrast, antiferromagnetic Kitaev exchange cannot be ruled out, but
requires a fine-tuning of all involved spin exchange couplings with only
moderate ratio $|K/J_1| \sim 1$. Our experimental data are incompatible with a
purely isotropic Heisenberg model.
