Here are photos of the plant.

Molly

That is Haworthia - a succulent, but not cacti. It needs more light - indicated by very light color and skinny leaves.

Is there a drainage hole in the container? If not, your potting soil could be staying too wet for too long. That could cause eaves becoming..."soft/almost transparent looking by the base"... How do you know if it is proper time to water and if the soil dries up? Easy and fast way to tell iss to use bamboo/wooden skewer, dowel, chopstick or even pencil. Insert deep into pot (3/4 deep or more) and leave for a minute or few. When pulled out, it is either dry and clean = time to water. If it is cold/damp and may have some soil particles sticking to it, soil is still wet and it shouldn't be watered yet.

Soil sold for cacti (and succulents) is often full of peat, and not easy to keep moist when needed. You can help it by getting a bag of perlite, and mixing with C&S soil in 1:1 ratio.

Thank you Rina, nicely done.

Hi there Rina!
The pot does not have a drain hole. I was instructed to put a layer of rock/small pebbles at the bottom then a layer of the cactus soil. I sprayed the plant with water trying my best not to over water, but now I'm questioning both of these tactics. It appears to be dying quickly too (bottom leaves are getting mushy), so i'm concerned if I need to repot/remove the pups or if I should wait? I'm not sure if the pictures do it justice. There's pups coming in on either side and one directly in the middle.

Those are typical instructions, but not best for the plants/succulents. Layer of rocks doesn't do much; if there is too much water, it just accumulates between the rocks, getting re-absorbed by soil and keeps it too wet. It is much better to get container with drainage hole. No layering - whatever ingredients are used for potting mix, should be mixed thoroughly.

If the plants leaves are too soft-mushy, plant is most likely rotting. If my plant, I would take it out asap; get off all the soil from the rootball. If there is any healthy part of plant, it could be re-rooted. But if there is any rotting matter left, it will, likely, spread and kill the plant. Set salvaged part on some newspaper or paper towels to soak up extra moisture. Take photo and post...you can leave it like that overnight or a day so plant dries up. Make a better draining mix and stick plant in it. Do not water for a while - succulents will survive for long time without extra water. keep it in good light - haworthias do not need strong sunlight but need good light. Some bottom heat could help if available (heat mat for example), but not absolutely necessary.

I found photo of some rotting leaves:

Soft/waterlogged, discolored (dark or watery-yellow) and falling off the plant very easily. These were removed from overwatered plant and discarded...

Here is photo of some rootless offsets/pups, and they were potted in a mix of grit and perlite. They have not been watered for about 3-4 weeks and are now rooted and potted separately (smallest one didn't make it :):

Hi Rina,
Wow!! Thank you so much!! This is exactly what I needed. Now, when mixing the perlite and cactus soil, is it about 50/50?

Yes, ppl using that kind of potting medium use 50:50. It could be even more of perlite. Here is photo one of our members, Crenda, posted before:

Oh wonderful! Thank you so much! :)