In this work, we formulate and rigorously test an alternative approach to go beyond the second order cumulant expansion. The method relies on applying a dampening factor to the third order cumulant correction that approximately accounts for the effect of neglected higher order cumulants in cancelling divergences in the third order lineshape. We demonstrate that the dampening factor Φ is a function of the non-Gaussian nature of the energy gap fluctuations quantified via the skewness and kurtosis of the distribution. This function Φ can be directly parameterized using the exactly solvable model system of the GBOM Hamiltonian. We show that the approach rigorously removes unphysical features from linear absorption spectra in both harmonic and anharmonic model systems, as well as realistic condensed phase systems sampled from MD. Additionally, the approach yields an excellent agreement with the FC method for the GBOM Hamiltonian including Duschinsky mode mixing and changes in PES curvature, thus opening up the possibility of accounting for these effects from first principles in simulations of molecules embedded in condensed phase environments.