Put the weight on the heel of your hand and higher.
I actually learned this from a personal trainer. He had me doing dumbbell curls and when the weight got higher, my wrists hurt all the time. So he asked me to instead of grabbing the weight, 'balance' it on the heel of my hand and hold it there with my fingertips (applies to hand facing up). It took a while to get used to, and the pain went away slowly, but this worked.
Fast forward to carrying kids all the time. While it really helps to be able to carry them longer if I've been keeping up my weight routines (whoever says bicep work is vanity was never a parent), and the same holds true.
Obviously you would never 'balance' your child on your palm. But when picking my children up, I have to 'choke up' towards the thumb, have my hands/forearms more perpendicular than parallel to the ground, and supinate my hands a bit to try to present the heel of the hand (and palm) to the child's armpits instead of the crevase of the thumb/forefinger. This requires bending down with my knees (OK now you have to keep your squat regimen up at the gym too) and getting closer to the child, all to relieve my wrists (when they are flaring up).
Then when carrying the child, by default (if it's a toddler - assuming that as a baby would be less likely to stress your wrist), you want to have the 'far bun' of the child in the palm of your hand. Instead of doing that, put the hell of your hand just past the far bun, with both buns resting on your forearm. (feel like we're working at a hamburger joint here, huh?)
Then let your hand extend past the child, almost limp, holding it up next to the child's body so you don't catch it on things as you walk around, and hurt it - I've done that.
Basically, the child is just sitting higher up on your arm (and maybe wrist) and less on your hand. I've noticed that none of my kids like this as much. Of course they don't like being practically dropped when pain randomly shoots through my wrist, either.