Carpet remnants are a great way to protect you --- and the cacti --- when moving them. Now is a good time to move them before the nights get cooler. Generally you want to dig a hole 2-3 times the width, but only to the depth of the rootball. Home Depot has a great palm/cactus soil mix. I blend it half and half with the native soil when transplanting these guys, seems to work well.

When I have transplanted cactus, I first wrap the cactus with a thick layer of foam pad to protect the thorns, and then wrap with a layer of carpet or movers blanket to protect me. It seems to me that the use of carpet alone is too hard on the cactus, and leaves broken and/or bent espinas.

The century plant is a different matter. I once impaled my thigh while transplanting one. Possibly covering the tips in the direct line of fire with wire nuts might work, though I don't know if they would damage the tips or not. It might be worth the experiment. Also, it may be necessary to let the root heal over if it is similar to that of the aloe. It has been so long ago that I transplanted the century plant that I've forgotten the type of root it had.

Terran

I don't know if you've already moved them or not, but if not, here's a website with step-by-step photos showing a good way to do it -
http://cals.arizona.edu/pubs/garden/az1376.pdf

And here's some info on transplanting -
http://cals.arizona.edu/yavapai/anr/hort/byg/archive/cactusculture.html

Hope it helps :)

Here is a link that might be useful: HOW TO TRANSPLANT A CACTUS