Answer: Sounds as though it's time to remove the little offshoots and pot them up, then discard the old, sick cactus. Just grasp each carefully and twist gently to remove the offshoot and all of its attached roots. Pot size will depend upon on how large the new little cacti are. Find one that will accommodate the roots without crowding.
Cactus do not like to sit in wet soil so buy a soil mix especially for cactus or mix your own by combining 1 part potting soil to 2 parts sharp sand, and 2 parts perlite or pumice. Put tiny pebbles on top of the soil after planting to help aid drainage.
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