Recent studies have shown that the observed color distributions of Type Ia supernovae (SNe Ia) can be well described by a combination of a dust distribution and an intrinsic color distribution. Using the Pantheon+ sample of 1701 SN Ia, we apply a new forward-modeling fitting method (Dust2Dust) to measure the parent dust and color distributions, including their dependence on host-galaxy mass. At each fit step, the SN Ia selection efficiency is determined from a large simulated sample that is reweighted to reflect the proposed distributions. We use five separate metrics to describe the goodness of fit: distribution of fitted light-curve color c, cosmological residual trends with c, cosmological residual scatter with c, fitted color-luminosity relationship β SALT2, and intrinsic scatter σ int. We present the results and the uncertainty in 12-dimensional space. Furthermore, we measure that the uncertainty on this modeling propagates to an upper threshold uncertainty in the equation of state of dark energy w of 0.014(1) for the Pantheon+ cosmology analysis and contributes negligible uncertainty to the Hubble constant H 0. The Dust2Dust code is made publicly available at https://github.com/djbrout/dustdriver.