Maybe it's not your finger. When I took my first guitar lesson, I told the prof that I had difficulty playing a C chord, and he said, "You're not supposed to play C on the guitar." I was liberated.
F is worse, so I suppose it applies even more.

Being a very good teacher, he did point out to me that the index finger can swivel on its base so that you can play strings with the side of it, not just the tip. Sometimes this helps. Nonetheless, I still believe that "you're not supposed to play C on the guitar."

If you are having to play lots of C chords to accompany a song, change the song to D.