We report a spin resonance study of the family of quasi-two-dimensional organic (super) conductors β″-(BEDT-TTF)4[(H3O)M(C2O4)3]S, where M is a 3d transition metal ion and S is a host solvent molecule. The spin systems for M=Cr3+ (S=3/2) and M=Fe3+ (S=5/2) are investigated by means of both resonant and field modulation techniques in the frequency range between 50 and 313 GHz. The role of the different solvent molecules in determining the degree of spin-orbit coupling and the local symmetry at the metal ion site is established. The low temperature behaviour of intensities, positions and widths of the resonant lines shows significant modifications of the spin-orbit coupling, and of the inter- and intra-ionic spin-spin interactions. Despite the onset of a weak antiferromagnetic internal field at low temperature, the ultimate narrowing of the lines suggests spin-lattice interactions may still be the dominant relaxation process. Diamagnetic screening in the mixed state of the superconducting samples for fields parallel to the quasitwo-dimensional layers induces additional lineshifts only below B=2.5 T and T=4 K, determining the threshold of full field penetration within the anion layers.