Fence gate sagging. Post twisting. Any ideas on how to solve this?
I am not sure what the cause of your post twisting is, although it doesn't look twisted in the picture. If you didn't cement the post in using postcrete or other quick drying cement, the soil packed around it can settle allowing the post to twist.
As for the gate sagging, it seems from the photo that you have not adequately braced the gate, so it will continue to sag more in time.  It may look square at first glance but to me there is a slight crookedness (look at the distances between the top left of the gate to the top frame, and the top right of the gate to the top frame).
Adding diagonal bracing will prevent the weight of the gate pulling down on the horizontal bracing at the part furthest from the hinges.  I would take the gate down, re-square it if aesthetically it would look better, and add diagonal bracing. Brace the top edge furthest from the hinges to the bottom of the hinged side using wood.  (In the photo of the bracing you have here, brace from top left to bottom right).
Downward forces furthest from the hinges will then be transferred to the bottom of the hinged part of the gate.
You could use wire but if you do, it won't be as strong on the bracing and the diagonal would need to be in the opposite direction.
The only thing is you said you trimmed the picket bottoms so if you re-square the gate it could look slightly crooked at the bottom, but that is much less noticeable and the gate won't sag any further.
Edit after further thought
Thinking about the size of the gate, which by the photos I would guess is a 1.8m (6ft) high gate, using wire for diagonal bracing may not be adequate and considering the weight of the downward forces which will be there, using wire would put a huge amount of strain on the top hinge which isn't that big looking at the photo.
I would use wood bracing in the manner described above.  It will not be as strenuous on the bottom hinge, and it will also lower the strain on the top hinge.