How you finger the F chord is not a LAW.. Your index finger can hit or rest on the D and G strings, as their sound is produced above your middle and ring finger. Some experimentation with your index finger such as rolling towards the side or moving verticle of the strings.
Your fingers do need to adjusts like your finger pads needed to form calluses.
Start with the index finger getting a clean sound then add the middle finger.. get the good sound then add the ring finger..
The clarity will come.. Maybe not everytime at first, but it will. mostly relax and and enjoy what is happening with your progress..
Just for fun, try the F with Middle Ring and Little Finger..
In the "CHORD FINDER" look at the A chord in the Second Variation and try that.
Hope this helps
Gordon