One thing I noticed about your process for computing the Fourier transform is, did you also include any “broadening”? The most common type of broadening is to multiply your time-dependent correlation function by e^{-\alpha |t|} for some value of \alpha which will have the effect of replacing any singularities in your spectral function by Lorentizians of width \alpha. It can specifically help with issues related to long time behaviors of correlation functions.