Low poly character eye modeling I have a problem with modeling :

This is my rough sketch of my character.
Now I applied the face png to the 3d model with &quot;Shrinkwrap modifier&quot;.
Result was devastating :

from this..
to this.
The eyes are distorted and bent so much from the original intended image as you can see :(
I've been searching for an answer to this on youtube but found none so far.
What other ways can I apply the face just like the first image at the top?
++++++ Below is a reference I used : 
seems like extrude function was used for this reference character's eyes but how can I select the sections that will look exactly like those eyes??
Thank you for reading this question! :)
Use Shrinkwrap &gt; method Project / Negative / Offset 0.01

Another try to flatten is to change camera Focal Length.
Example - from original 50 to 180 mm.