Exhibiting rich magnetic behaviour and potentially multiferroic properties, the dugganites, a Te(6+) containing subgroup of the langasite series, are an attractive family of compounds for future study. It was recently shown that Pb-bearing members of the dugganite series undergo distortions away from the P321 symmetry that is characteristic of the langasites. Here, we detail the consequences these distortions have on the magnetism exhibited by Pb3TeCo3V2O14 and Pb3TeCo3P2O14, solving the magnetic structures of both compounds with respect to a new supercell. Using neutron scattering and magnetic susceptibility measurements, we show that small applied magnetic fields can seriously perturb the delicate magnetic states in both of these systems. This is further demonstrated by presenting how doping P(5+) onto the nonmagnetic V(5+) site completely changes the magnetic structure from either of the end series members. Finally, it is shown using inelastic neutron scattering and magnetic susceptibility measurements that Pb3TeCo3V2O14 can be characterized using a model for isosceles trimers, which do not exist in the previously reported P321 subcell.