I'm not seeing any rot in your pics. Maybe you didn't include that in your pics or you've misidentified something as rot which really isn't. I'm not sure. I am seeing corking and shriveling because of underwatering but looks like you have an Opuntia which often look shriveled this time of the year if kept outside. Corking is a browning of the epidermis of the cactus skin which can be caused by culture conditions that aren't right for the cactus or age of the cactus. The new growth is thin and etiolated because the cactus is not getting enough sunlight. Opuntia can need a lot of sunlight to grow correctly. If you increase the sunlight then do it gradually or else the plant can sunburn. Looks like your cutting isn't callousing because the part you cut off was too dehydrated before it was cut off.

Also, I'm not sure why the plant's root ball is raised. Seems like the plant should have been planted deeper. The roots at the top are exposed but Opuntia are often tough so I'm not sure how much it matters.

Hi, thank you so much for your help. It is very possible I misidentified it as root rot - I am a beginner plant owner, and admittedly only decided it was root rot because I have a history of overwatering plants. Looks like I went too far the other way this time.

I repotted it into a pot that has a drainage hole at the bottom, so I can water it more often without having to worry about whether water is pooling at the bottom. I also buried the root ball this time. I will also slowly introduce it to more light.

Thank you so much again for your help!