I have one. Tephrocactus geometricus. It had no roots and a small black spot on it when I got it but it (the spot) has not gotten any larger. My understanding is that some blemishing on these is not uncommon.

Mine is in pure pumice (so water runs right through) in full sun. I water every 10-14 days or so just depending on the weather, cloud cover, etc.

It is also my understanding that these tend to be rot prone, but as I am still learning and we have different plants I hesitate to say what is going on with yours. My initial feeling though is that your medium retains too much water and since it sounds like they don’t get a whole lot of direct sunlight, you are overwatering by watering them once a week.