A theoretical study is presented of phonon scattering due to localized vibronic interaction using the method of double-time thermal Green's functions and the $T$-matrix formalism. The impurities considered are Jahn-Teller ions with orbital doublet electronic states in dilute concentration and their coupling to the continuum of lattice modes is assumed to be linear and strong. A random-phase-approximation decoupling of the Green's-functions equations is performed after a transformation of the phonon coordinates is carried out, which separates out one strongly interacting set of modes and leaves the rest weakly coupled. The features of the low-frequency resonances, centered about the shifted rotational excitations of the quasimolecular Jahn-Teller system (electronic states plus the set of interacting modes), are investigated and lifetime broadening mechanisms of these excitations at different temperature regions are discussed. The phonon scattering rates calculated by us are applicable to low-temperature thermal-conductivity and acoustic-attenuation measurements and are compared with thermal-conductivity data for ${\mathrm{Cr}}^{2+}$ in MgO and ${\mathrm{Ni}}^{3+}$ in ${\mathrm{Al}}_{2}$${\mathrm{O}}_{3}$.