Li3Co2SbO6 is prepared by molten salt ion exchange and its structure refined by the Rietveld method confirming the honeycomb-type Co/Sb ordering of its Na precursor. Monoclinic rather than trigonal symmetry of Na3Co2SbO6 is directly demonstrated for the first time by peak splitting in the high-resolution synchrotron XRD pattern. The long-range antiferromagnetic order is established at TN ≈ 6.7 K and 9.9 K in Na3Co2SbO6 and Li3Co2SbO6, respectively, confirmed by both the magnetic susceptibility and specific heat. Spin-wave analysis of specific heat data indicates the presence of 3D AFM magnons in Na3Co2SbO6 and 2D AFM magnons in Li3Co2SbO6. The field dependence of the magnetization almost reaches saturation in moderate magnetic fields up to 9 T and demonstrates characteristic features of magnetic field induced spin-reorientation transitions for both A3Co2SbO6 (A = Na, Li). Overall thermodynamic studies show that the magnetic properties of both compounds are very sensitive to an external magnetic field, thus predicting a non-trivial ground state with a rich magnetic phase diagram. The ground state spin configuration of Li3Co2SbO6 has been determined by low-temperature neutron powder diffraction. It represents a ferromagnetic arrangement of moments in the honeycomb layers with antiferromagnetic coupling between adjacent layers.