Exact (non-Born-Oppenheimer) numerical solutions of the time-dependent Schrodinger equation for a I-D model of the one-electron isotopomer systems H-2(+), HD+, and HT+ have been obtained in a two-color laser control scheme, w + 2w, of dissociation and dissociative ionization in the high-intensity (I greater than or equal to 10(13) W/cm(2)), nonlinear, nonperturbative regime of laser-molecule interaction. Calculations have been performed at the fundamental wavelengths lambda(CO2) = 10.3 mum and lambda(YAG) = 1.064 mum in combination with their second harmonies in order to study the effect of laser frequency and permanent dipole moments on the electron-nuclear dynamics in the presence of periodic asymmetric fields. It is found that asymmetries occur in both the ionization and the dissociation. The phase sensitivity is shown to be quite different in the two wavelength regimes. Adiabatic and nonadiabatic quasistatic models of the dissociation and ionization processes are shown to offer simple interpretation of the calculated asymmetries in terms of tunneling ionization and barrier suppression dissociation.