Hi Andrija,
Pressing too hard is certainly a thing to examine.
For the fingertip sensitivity:
- I have trouble when I cut my nails too short. I need them fairly short so I don’t hit the fretboard before getting the string past the fret, so it is a balance.
- Do you use hand lotion that keeps your fingertips from hardening? There is a distinct difference between my picking and fretting hand fingertip rigidity.
- Do you regularly practice just after having your hands in water (shower, dishes, etc)? This softens up my fingers quite a lot for a short time.
Related to pressing too hard, pay attention to using muscles you do not need to get the strings pressed. For instance, fret your chord and stop there.
- Examine what muscles are in use and how tensely.
- Are you tightening your wrist where it doesn’t need it?
- Are you tense at the elbow without need?
- Are you tightening muscles on the opposite side of your hand that can be relaxed (they are just fighting the ones you need to squeeze the string down!)?
My wife tells me I make funny faces - muscles I don’t need to be using to play! 
Hope that gives you something to help!