We examine the strong-field dissociation behavior of diatomic molecules under two distinctive physical scenarios. In the first scenario, we discuss the dissociation of the isolated hydrogen and deuterium molecular ions. The dynamics of above-threshold dissociation (ATD) are investigated over a wide range of green and infrared intensities and compared to a dressed- state model. The second situation arises when strong-field neutral dissociation is followed by ionization of the atomic fragments. The study results in a direct measure of the atomic fragment's ac-Stark shift by observing the intensity-dependent shifts in the electron or nuclear fragment kinetic energy.