Self-interacting dark matter (SIDM) is one dark matter (DM) model that has been proposed to solve some of the outstanding discrepancies in observations of galaxies and what our current, cold dark matter (CDM) model indicate in simulations. While many previous works use dark matter only (DMO) simulations, we additionally consider the effects of baryons in full, hydrodynamic cosmological simulations. We investigated the morphology of five different early-type galaxies selected from the Illustris simulation and re-simulated with the IllustrisTNG model for baryons in the context of both CDM and SIDM cosmological paradigms. As a measure of morphology, we calculated the ratio of semi-minor to semi-major axis of the 3D (2D) ellipsoidal (elliptical-projection) particle distribution. In concordance with previous works, we find on average the central regions of haloes in our sample are rounder, for both SIDM and CDM models, than the outer regions. With the DM+baryons case, we find only slight evidence indicating SIDM models result in rounder matter distributions than seen in CDM models. However, the DM+baryons case is consistently rounder than the DMO case, regardless of cosmological model. While the inclusion of baryons may provide more robust estimates of halo morphology, perhaps providing better agreement with observations, this preliminary study indicates that one cannot necessarily use observations of halo morphology to solely distinguish between CDM and SIDM models.