How do I divide an overgrown cactus (looks similar to Mammillaria vetula subsp. gracilis Preiff)?
Very carefully so you don't get stuck. I have some kitchen tongs I use for things like that. I work the tongs in between two of the off shoots and then if they don't come apart from that I take a sharp knife and cut off an off shoot.
Then let it dry overnight and pot up in fresh cactus soil, something that is gritty and sandy, that drains well. I have re-potted lots of cacti.