Here's a photo of the roots.

While not the most ideal time to repot cacti, I would do it. Make sure you get them into a gritty mix and get as much of the old "dirt" off them as possible before your repot. The mold you see on the roots is probably actually a bunch of micro roots developing.
No, don't water before the move. Get clay pots an inch or two bigger than what they are in now. When you repot, let sit for about a week. You could then water once, making sure the water drains completely. I would not water again maybe until the end of January or February, depending on whether these are being kept indoors or out. Others will chime in with their 2 cents.

Nancy

There's no real way to know when they were last watered so If the roots are bone dry don't be afraid. Succulents need water too.

While you will hear dire warnings about transplanting in Winter, it really depends on your particular growing conditions. While most all of my tropicals and subtropicals go outside for the growing season, my cacti sit in an unobstructed, southern exposure where they get direct sun in the Winter but hardy any in the Summer due to the high sun angle. I have not had problems with Winter transplanting. I do not use a "gritty mix" per se but do use soils with coarse sand and incorporated perlite. I might water once a month in Midwinter (about the same for the Summer due to fact that they are more shaded in Summer). Personally, I might loosen up the root ball some, but not remove all the old soil as this seems likely to shock the plant. The only reason for removing most of the old soil would be if it were being planted into dramatically different soil type (such as a gritty mix).

Get some clay pots with holes in the bottom, a little wider than the cactus. I just buy commercial cactus mix, but others here may not approve. Be very careful, wear gloves and handle with thick strips of newspaper. Some of those tiny, almost invisible spines can be really tough to get out of a finger. Those look nice. Have fun.

You guys have been very helpful! If I decided on the gritty mix, how often do I need to fertilize them? I keep reading 1/4th strength of almost any plant fertilizer.

I would go with every other or every third time you water in the growing months. You won't need to do this when they are dormant.

Nancy