Formulas are obtained in the paper for the instantaneous translational temperature changes and for the average number of vibrational quanta at the vibrational relaxation of nonequilibriumly excited anharmonic Morse oscillators under nonisothermal conditions in presence of the dissociation. The formulas generalize the known formula by Losev, Shatalov, and Yalovik (1970) and in contrast to the formulas by Gordiets et al., obtained for the time intervals of t.tVT, they allow one to calculate the relaxation time within the interval of t<tVT. The expressions obtained explicitly show the dependence of the relaxation rate on the distribution function, degree of excitation and on the nonisothermicity factor and anharmonicity as well. Relative contributions of all the nonlinear factors considered are illustrated with the results of numerical calculations.