Nick,
Sounds like you are doing everything right. Is the organic mulch you mention placed on top of the soil or mixed in with the cactus mix, potting soil, and perlite? If so, it may be retaining too much moisture for that plant. I would try removing it. Conversely, I have had rooted plants become soft and sometimes wrinkle when they actually needed water. After I watered them, they firmed up in a day or two. I, too, thought I had stem rot, but they are growing nicely now.

Some AZ people use shade cloth to filter the sunlight, but others don't and both seem to be successful. It might be sunburn, so you may want to shield those plants, or move them to more indirect sunlight.

One person told me to use a needle to poke a small hole where the plumie was shriveled to see if latex oozed out. I tried that on several plants and all oozed latex. Only one of those eventually had stem rot, so it may be a good test, but I don't have enough experience to confirm it.

Good luck,
Mike