Is there a way to move the selected face on a curved mesh without getting the mesh distorted? I need to be able to reposition the face within the area.
Change your transform orientation to normal g+x or g+y or using the transform widget will give you the transform you want for this mesh. If it has more geometry, being truly curved, put your 3d cursor in the center and rotate the geometry with the 3D cursor set as your transform.
Using shrink-wrap or snapping to another object can give some distortion.