In this work, we describe how the energies obtained in molecular calculations performed without assuming the Born-Oppenheimer (BO) approximation can be augmented with corrections accounting for the leading relativistic effects. Unlike the conventional BO approach, where these effects only concern the relativistic interactions between the electrons, the non-BO approach also accounts for the relativistic effects due to the nuclei and due to the coupling of the coupled electron-nucleus motion. In the numerical sections, the results obtained with the two approaches are compared. The first comparison concerns the dissociation energies of the two-electron isotopologues of the H, molecule, H-2, HD, D-2, T-2, and the HeH+ ion. The comparison shows that, as expected, the differences in the relativistic contributions obtained with the two approaches increase as the nuclei become lighter. The second comparison concerns the relativistic corrections to all 23 pure vibrational states of the HD+ ion. An interesting charge asymmetry caused by the nonadiabatic electron-nucleus interaction appears in this system, and this effect significantly increases with the vibration excitation. The comparison of the non-BO results with the results obtained with the conventional BO approach, which in the lowest order does not describe the charge-asymmetry effect, reveals how this effect affects the values of the relativistic corrections.