How do galaxies transform from blue, star-forming spirals to red, quiescent early-type galaxies? To answer this question, we analyzed a set of 26 gas-rich, shocked post-starburst galaxies with Hubble Space Telescope (HST) imaging in B, I, and H bands and Sloan Digital Sky Survey (SDSS) i-band imaging of similar depth but lower resolution. We found that post-starbursts in our sample have intermediate morphologies between disk- and bulge-dominated (Sersic n=1.7_-0.0_^+0.3^) and have red bulges, likely due to dust obscuration in the cores. A majority of galaxies in our sample are more morphologically disturbed than regular galaxies (88%, corresponding to >3{sigma} significance) when observed with HST, with asymmetry and Sersic residual flux fraction being the most successful measures of disturbance. Most disturbances are undetected at the lower resolution of SDSS imaging. Although ~27% galaxies are clear merger remnants, we found that disturbances in another ~30% of the sample are internal, caused by small-scale perturbations or dust substructures rather than tidal features, and require high-resolution imaging to detect. We found 2.8{sigma} evidence that asymmetry features fade on timescales ~200Myr, and may vanish entirely after ~750Myr, so we do not rule out a possible merger origin of all post-starbursts given that asymmetric features may have already faded. This work highlights the importance of small-scale disturbances, detected only in high-resolution imaging, in understanding structural evolution of transitioning galaxies.