I got a Crosby's Prolific aloe and I noticed two pups growing beside the main plant. The pups are connected by a strong thick root and I'm not sure if I should just pull it. Any advice would be appreciated!

I think the pups are actually branches off the main stem, so you would have to cut them free from the main stem. I would take sharp pruners and cut straight across the base of the offset right where it comes off the main stem. Don't pull, cut. The offset on the right (in the first picture) looks like it will come with its own roots (remove carefully) if you go that route, the one on the left may not have any roots. Cuttings should root quickly (within about 3 months starting from a rootless cut, I would expect).

Those pink flowers on your plant are not a match for the CP flowers that I'm familiar with, and the surfaces of the leaves are not spiny on the undersides, unlike the CP I know:



So I would guess you have something different, but very similar-looking.

When you separate the offsets, give them time to heal. Maybe pot them up in dry soil and then wait a week or so to water. If you end up with a rootless cutting, leave it in a place with good ventilation for a couple of days to heal before potting it up. The important thing is not to water immediately after potting up either mother or daughter plants... wait a week or so first.

I started cutting the right offset and it's a bit loose now, but I found one of its roots leading deep into the main root system. Should I cut the root off?








Also, I noticed the right offset's roots are pink and yellow. Is this normal?
Thank you!

The roots look good to me. Try gently wiggling the offset around, maybe poke your finger into the root ball to help the attached root get free of the other plant. It would be best if you could get the offset with a root, but like I said earlier, that is not a requirement, it simply saves you a slight delay to the process.

I managed to get the right offset off, but the root was really anchored in there. I tugged at it to try to get the root out but instead the whole offset came off. So now the offset has a cut root. Can I pot it right away or should I let it dry? Should I let the main plant dry too since both of them have a cut?
The right offset:



The main plant:

Pot them up in dry soil and wait a week to water.

Got it! How should I cut off the left offset? It feels like it's whole stem is connected to the main plant when I wiggle it. Should I just cut where the offset's stem and the main plant's stem meets and leave a big cut? It has one root of its own off to the side.

Yes, cut where they meet, straight across the stem of the offset.

I cut the offset where they met. The offset and the main plant both have a big white cut where I cut them. Should I pot this offset too or should I wait for the big cut to heal? Will the main plant be alright with such a big cut in soil? Thank you!
The left offset:



The main plant:

You can wait if you want. It shouldn't really matter.

Okay, I potted the main plant and the right offset. Thank you for all the guidance!