Iride01 

We must be someone else.

I'd push them further away from you on to the curved part of the bar until the body of the STI is almost but not quite horizontal. And don't just accept that the current angle your STI's are toed in or out is great. Change it up and actually find out for yourself. And don't just do a ride around the block and say it feels odd. Try it out for several good 90 minute or more rides.

metropical

okey dokes.

cyclezen

I'll offer some other options...
The Stem rotated to the 'UP' angle not only makes you a bit more upright, but also moves the bars back towards you. There's a fine balance to that. Having too little reach actually forces more weight to the bars, which now is jammed into the brifters thru the Y between the thumb and remainder of hand, bends the wrist and forces weight into the Ulnar channel - everything you describe.
Having the Brifters up and back on the bars creates MORE of this weight driving into the brifters/bars/
I would suggest leveling the bars
Move the Brifters FORWARD, closer to the bend, until the main lower body of the brifter is inline/parallel to the bar top (as in Image of CAAD9 below) - NOT angled up and back as is now.

2009- CAAD 9

The stem can remain rotated to the + angle...
This will add a bit of reach, but also allow the hands to cradle the 'front' of the brifters (as designed). This bit of extra reach will allow the weight to be distributed over the greater length of the hand.
a straighter wrist will allow you to grip around the brifter base, without cocking the wrist. The levers will be easier to grasp. Shimano has spend plenty of effort, over the years, to prefect this design.
IF, when doing the above, you're still having a lot of weight on the hands, bend the elbows a bit, when riding.
Drop/pull down the shoulders away from the ears. If in a gear which you have to pressure a bit, you should have a light touch on the bars, easily moving the hands around WHILE PEDALING.
If you can;t do that, your BODY COG is prolly a bit too far forward.
Seat
If all the above still exists - you might have the saddle a bit too forward. Mark the back of saddle rail with marker or a bit of tape, where the seat clamp stops. Then move the seat back 5mm and level. Ride 4 or 5 times before deciding anything. If that helps a bit, move another 5 mm back.... If the saddle begins to feel 'High', you can drop saddle height 2mm or so
We're talkin millimeters here, NOT inches...
Ride for 10-12 rides, then decide...
Ride On
Yuri

metropical

Thanks Yuri. I'll level the bar and move the Brifters into the curve as you describe.
And try some seat movement.
As I said, hoping the Trek store that I have been rotating the saddle from, has stems I can try as well as narrower saddles I can try.
Does seem to me going from a 125mm to a 165 is a big jump.
Measuring the sitting seat bone doesn't seem to take into account the actual cockpit geometry.
As others have pointed out, a 58cm might be a better choice for me, but I'm not about to drop $1200 or more right now.
And I'd think with a little care the CAAD can still be good. It is, after all, the last of a generation.

SimicRecluse

Gloves, and foam padding for hand comfort. There are nice high density foam options for grips off of ebay and amazon. No reason to suffer on bar tape with little padding

SoSmellyAir

The angle of the STI lever bodies in this photo is much closer to ideal than that on metropical's above photo. Note the angle of the ramp, with reference to: Drop bar hand positions | BikeGremlin.

metropical

I ended up trying a couple saddles based on seat bone measurement. But coming from a 125 to a 175, then a 165 was making things worse.
Long story short, a couple of stem changes I ended up at 145mm Bontrager. with the OEM stem, 7º or whatever it is. So far so good.
Hand numbing, so far, is gone. I don't recall if they adjusted the STI to the bar.
Still need to retape the bar.