Converged quantum mechanical vibrational-rotational partition functions and free energies are calculated using realistic potential energy surfaces for several chalcogen dihydrides (H2O, D2O, H2S, H2Se) over a wide range of temperatures (600-4000 K). We employ an adaptively optimized Monte Carlo integration scheme for computing vibrational-rotational partition functions by the Fourier path-integral method. The partition functions and free energies calculated in this way are compared to approximate calculations that assume the separation of vibrational motions from rotational motions. In the approximate calculations, rotations are treated as those of a classical rigid rotator, and vibrations are treated by perturbation theory methods or by the harmonic oscillator model. We find that the perturbation theory treatments yield molecular partition functions which agree closely overall (within ∼7%) with the fully coupled accurate calculations, and these treatments reduce the errors by about a factor of 2 compared to the independent-mode harmonic oscillator model (with errors of ∼16%). These calculations indicate that vibrational anharmonicity and mode-mode coupling effects are significant, but that they may be treated with useful accuracy by perturbation theory for these molecules. The quantal free energies for gaseous water agree well with previously available approximate values for this well studied molecule, and similarly accurate values are also presented for the less well studied D2O, H2S, and H2Se.