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The magnetic properties of the two isostructural molecule-based magnets, Ni(NCS)$_{2}$(thiourea)$_{2}$, $S$ = 1, [thiourea = SC(NH$_2$)$_2$] and Co(NCS)$_{2}$(thiourea)$_{2}$, $S$ = 3/2, are characterised using several techniques in order to rationalise their relationship with structural parameters and ascertain magnetic changes caused by substitution of the spin. Zero-field heat capacity and muon-spin relaxation measurements reveal low-temperature long-range ordering in both compounds, in addition to Ising-like ($D < 0$) single-ion anisotropy ($D_{\rm{Co}} \sim$ -100 K, $D_{\rm{Ni}} \sim$ -10 K). Crystal and electronic structure, combined with DC-field magnetometry, affirm highly quasi-one-dimensional behaviour, with ferromagnetic intrachain exchange interactions $J_{\rm{Co}}\approx+4$ K and $J_{\rm{Ni}}\sim+100$ K and weak antiferromagnetic interchain exchange, on the order of $J'$ $\sim-0.1$ K. Electron charge and spin-density mapping reveals through-space exchange as a mechanism to explain the large discrepancy in $J$-values despite, from a structural perspective, the highly similar exchange pathways in both materials. Both species can be compared to the similar compounds $M$Cl$_2$(thiourea)$_4$, $M$ = Ni(II) (DTN) and Co(II) (DTC), where DTN is know to harbour two magnetic field-induced quantum critical points. Direct comparison of DTN and DTC with the compounds studied here shows that substituting the halide Cl$^-$ ion, for the NCS$^-$ ion, results in a dramatic change in both the structural and magnetic properties., Main text: 13 pages - 11 figures, 2 tables. Supplemental information: 6 pages - 2 figures, 6 tables