A polyatomic gas with slow relaxation of internal modes is considered, and the Navier-Stokes equations with two temperatures, the translational and internal temperatures, are derived for a gas with constant specific heats formally, by the Chapman-Enskog procedure on the basis of the ellipsoidal-statistical (ES) model of the Boltzmann equation. The Navier-Stokes equations are then extended to a gas with temperature-dependent specific heats. As an application, the problem of shock-wave structure is investigated numerically using the extended equations. It is shown that the two-temperature Navier-Stokes equations describe the shock-wave structure correctly even for a moderately strong shock wave.