I rescued this baby a few weeks ago and within days the bottom part began to get very soft and slimy so I put it outside in the shade and stopped watering it. The bottom has dried and seems to be holding its own. Should I cut it off and try to re-root it? Also The yellow part has produced “babies” should I be doing something with them or just leave them alone?
What a beautiful cactus! It looks like the top part, which has been grafted on, can be saved, but it's possible the bottom may not survive.
If the bottom is dried out and seems to be doing ok, I'd let it continue to dry for another week or so and then begin watering again, following this technique.
However, if things seem to decline in the next week or two, you'll likely want to regraft the yellow section of the cactus to a new base.
The “stem” of your cactus looks like Hylocereus. You can purchase another, and graft the top bulb (which appears to be a Gymnocalycium mihanovichii) to it.
The top “bulbs” are unable to survive on their own, as they are unable to make their own chlorophyll — this is how they have such vibrant colors. Any babies removed from the top section will need to be grafted onto another stem in order to obtain nutrients.
To graft your cactus, you'll need the following:
A sharp, sterilized knife
A new pot
I would suggest watching this video, and following the directions in this post.
Good luck, and happy grafting!