Why are my wrists the first part of my body to start aching during a long ride?
As @Daniel has commented, and I agreed, you probably have a bike fit problem. We cannot be sure; the only way to find out for sure is to get a professional fitting done. Having said that there are still a few things you can try.


  While you say you try to adjust my hand positions, what positions do you use most?
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  The Allez handlebar setup is one I just would not be comfortable with: not enough drop, the straight section of the drop is too low an angle, and the hoods are too low.
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  @andy256 mostly on the hoods, I don't tend to use drops very often - only for variety. 


There are several different ways of riding on the hoods. Try them out if you haven't already:


With the thumbs only on the hoods, palms vertical, fingers pointing forward touching the brifter.
You can easily operate brakes from this position.
With the thumbs only on the hoods, palms vertical, wrists high, two fingers wrapped under the hoods, similar to sprinting on the hoods.
You can operate brakes from this position, but not so easily.
With the base of the thumb on the hoods, palms near 45°, one or two fingers touching the brifter.
This is a common position, but it IMO it is the one that causes such problems.
You can easily operate brakes from this position.
Palms further back, with some fingers either side of the "horn" of the brifter.
You can not easily operate brakes from this position.
Good for variety. Can alleviate discomfort.
Palms further forward, resting on the "horn". Fingers forward, just dangling, palm horizontal.
You can not easily operate brakes from this position.
Good for variety, and gives more stretch.


The last two are clearly just for variety. They have obvious dangers.

There are other positions you can use for variety. 


Hands at the end of the drops, arms almost straight. Your body position is not much different from having your hands on the hoods. You can't reach the brakes.
Hands further forward on the drops, just touching the upward curve. This requires a different wrist position, but may be worse for you. Some people can reach the brakes, but on the Allez I don't think you will.
Hands on the tops with elbows bent and tucked back beside your body. This can be uncomfortable at first but may grow on you. Your body position is still low.
Again, you're not anywhere near you brakes.
Sitting up more, with hands on the tops. Your hands can be close to the middle or wider to the corners. You'll find each position puts your wrist in a slightly different position.
Again, sitting up more, with hands on the corners. You can have your fingers pointing forward and to the inside, or have your thumbs on the bars with fingers underneath. These have very different wrist positions.
And finally, if you're really into relaxed positions and don't mind a little less control, rest your wrists on the tops, with hands dangling.
There are obvious dangers in this position.


These positions are not for everybody. Some don't work on a given bike, or with a given person's body. Bikes and people are all different. Try some of them and see if they help.

As with any on bike experiments, do it in a safe location.

I have not mentioned adjusting the brifter position. On many bikes I'd try that, but I don't think that would work on the Allez. But you could rotate the bars slightly to make the hoods angle up more. 5 to 10° should be enough to have an effect. I think changing the angle is more important than lifting them (everybody has their theory).

Finally, try taking one hand off the bars and shake it from time to time, then do it with the other hand.