I've been researching into this for the past year, yet I have only recently tried to seriously attempt a character like this. Last year, Land of the Lustrous was released, and so I wanted to try out that style as well. I've done research for the past few months on whatever was released on the CGWORLD JP website (I don't speak Japanese) and attempted to recreate it. Keep in mind that I did this all in Maya while studio Orange did it all in 3DS Max with in house tools.
NORMAL/UNEDITED


Now, your first issue with this may have something to do with the eyes. Yes, that eye is floating because it is a separate mesh. I know it looks weird in the angle shot, but that's how studio Orange does it. In the next pictures, you'll see why.
EDITED


According to what I've researched, studio Orange has an in house tool that automatically deforms the face based on the camera angle. In this case, there would be an "angled camera view right" blendshape for the face and eyes. While I didn't use a blendshape for this example, I was only testing it at the moment to see if it would work.

Next up, I'm going to try doing the mouth in a different style. The models have a line floating in front of the faces in order to simulate the line of the lips as you get to a certain angle of the face. From there, they draw the mouth when they are talking in manually using a 2D program. I'm going to try something different.
Resources:
https://cgworld.jp/feature/201710-cgw231T2-houseki.html