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This is caused by fungus. Treat the plant with a fungucide and when you water, try to keep the water off the leaves. This encourages fungus to grow.

Pick a height you want it to grow back from and make a clean cut. For instance, you could top off the tallest trunks and pot them up to make new plants. Prune it up 10 - 12". The new sprouts will form at the top of the cut giving you another dracaena, depending on how many sprout.

It is a non-poisonous plant. It is safe in your classroom.

It sounds like it may have a fungus or root rot.

Check the roots and see if they seem mushy. If it has root rot, trimming it back would have reduced the load on the damaged roots and would improve the health of the plant. If it is root rot, make sure you only water when the soil feels dry to the touch and this article will help:
https://www.gardeningknowhow.com/plant-problems/disease/treating-root-rot-gardening-tips-for-housplants.htm

It may also be a fungus attacking the plant. Again, cutting off the infected parts can help because it removes some of the disease, but you can't cut parts off forever, right? :) So, also treat the plant with a fungicide to clear up any possible fungus.

It sounds like root rot has developed. Root rot is normally caused by over watering. Most plants, especially yucca, only need watering once the soil has dried. This article will help determine if this is indeed the problem and what you can do about it: https://www.gardeningknowhow.com/plant-problems/disease/treating-root-rot-gardening-tips-for-housplants.htm

You have a Draceana, aka corn plant. Here is more info: https://www.gardeningknowhow.com/houseplants/dracaena/dracaena-houseplant-care.htm