I don't know if CT will answer this, but I prefer overhead pressing with an open grip, with the bar resting on the heel of your hand... that pad at the base of your palm. The wrists are probably only back 45-55 degrees or so, just enough to get the bar directly over the forearm.
Obviously the lift can be done either way, with the wrists cocked backwards and the bar not directly over the forearm. However, most of the great pressers in history (i.e., before the clean and press was removed as an Olympic lift) used the form as I described.
If you want to dig a bit, I'd take a look at the writing of Charles A Smith, who wrote extensively on pressing back in the day.
I would venture a guess that with both the press and the dips, your wrist pain is coming from letting the wrists flex too much. As far as fixing them, fish oil + time... and then stop letting the wrist flex so much when you start back up again. It'll probably take time to get used to and the heel of your hands will probably be sore and feel bruised for a bit. After that builds up, you should be fine.