Cavity optomechanics aims to establish optical control over vibrations of nanoscale mechanical systems, to heat, cool or to drive them toward coherent, or nonclassical states. This field was recently extended to encompass molecular optomechanics: the dynamics of THz molecular vibrations coupled to the optical fields of lossy cavities via Raman transitions. The molecular platform should also prove suitable for demonstrating more sophisticated optomechanical effects, including engineering of nonclassical mechanical states, or inducing coherent molecular vibrations. We propose two schemes for implementing these effects, exploiting the strong intrinsic anharmonicities of molecular vibrations. First, to prepare a nonclassical mechanical state, we propose an incoherent analogue of the mechanical blockade, in which the molecular anharmonicity and optical response of hybrid cavities isolate the two lowest-energy vibrational states. Secondly, we show that for a strongly driven optomechanical system, the anharmonicity can suppress the mechanical amplification, shifting and reshaping the onset of coherent mechanical oscillations. Our estimates indicate that both effects should be within reach of existing platforms for Surface Enhanced Raman Scattering.