Soft and squishy is most often sign of rot (according to what you said). It also looks like it was potted in water-retaining soil, and too deep. I am judging that from soil visible in photo. And depth of potting from light area that is too far from a root ball.

Pups are not killing your plant; it is most likely wrong growing conditions = water-retaining soil. Many cacti grow many pups all over, and they do not cause any harm to mother plant.

Did plant get properly acclimatized to more sun?

Do you think I can save my main plant? I’ll get better cactus soil tomorrow! Right now he is out of the soil/pot.
And yes he was acclimatized slowly to more slowly.

It looks like no cactus I have seen before. I am wondering if it may be an etoliated Mammillaria. You say it is getting lots of sun but is that only recently? It looks like it was getting insufficient sun for the last year and that is why it is growing long and squishy. The roots look good and there is no sign of rot in the body of the plant. Repot it and leave the pups on. In full sun, they will grow normal and take over the shape of the plant. The main stem is not going to grow a good shape now and can be cut off once the pups grow in size (if it is a globular cactus as i suspect).