I hope that 3rd finger on the 6th string is a typo. The C note or root is on the 3rd fret 5th string. Sometimes the G note is also played on the sixth string 3rd fret together with the root note on the 5th fret, but not usually instead of the C on the 5th.
Try angling the headstock of your guitar more towards the front of your body like you were trying to point yor guitar to the wall in front of you. That will cit down the angle on your wrist. Tuck your elbow in to your side as much as possible and let your hand
slide futher around the neck moving your fingers closer to the 6th string. That might help if your not already doing all that. It surprises me that your struggling more with the C chord than the barr F chord. Most people find that one the toughest to get. Don't forget to shake your wrist out every ten minutes or so. Hope some of this helps.