Properties of a new thawed Gaussian propagator, recently suggested by Baranger et al. for motion under a smoothed or averaged Hamiltonian, are examined for scaled Morse and quartic oscillators. The resulting power spectra are similar to those determined by Heller's thawed Gaussian wavepackets governed by the classical Hamiltonian, but markedly less accurate than the power spectra obtained by Herman-Kluk propagation. A phase modification to the new propagator implies a new semiclassical quantization formula, which is shown to be analytically equivalent to the Bohr-Sommerfeld form for the Morse oscillator, for parameter values appropriate to molecular systems. Small differences between the two forms for the quartic oscillator are found to depend on the width of the coherent state used for the smoothing.