@dirge0matic Okay, I already told you what is distorting your mesh. What you can try is deleting the Subsurf, applying the Shrinkwrap, re-adding the Subsurf and then re-adding the Shrinkwrap. This might give a better result, because the mesh you are then shrinkwrapping is already closer to the target. You can also experiment with other shrinkwrap modes; I find that very often (but not always) Target Normal Project works a bit better than Nearest Surface Point.
