I land a slightly awkward punch and it causes a lot of wrist pain.
That's the key of course. You are probably trying to throw powerful punches instead of throwing
proper punches. Powerful strikes come only from proper technique. "Awkward punches" are weak and hurt you more than the other person. Take your time and practice proper technique.
Of course hand wrapping is
very important. It's also something that takes a little time to get good at. Just like a kid learning to tie his shoes. Be sure to use the 180" wraps, not the short ones. Be sure that the wraps have a little stretch to them as well. My favorite ones are sold right at Dick's (and I have tried a few). The Everlast brand, but again, be sure they are 180". A friend of mine goes to Thailand every year to train and showed me how they taught him to wrap his hands, which became my way forever after. It's the same method as going between the fingers, etc. except instead of putting my thumb through the loop, I fold a couple layers of material to pad my knuckles.
I had carpel tunnel before I started training and boxing got rid of my symptoms. I stopped training for a couple years, and my carpel tunnel came back, along with an elbow and shoulder injury (my job is very prone to these injuries). I started boxing training again, and once again, the pain and injuries went away. So I can't say if this is the same for you, but in my experience this kind of training will prevent these kinds of injuries in the long run, not make them worse. Unless of course, you throw a bad punch or neglect handwrapping. That would hurt anyone.