Thank you for your question. What you don't mention--and what I suspect is the problem--is that the plant needs to be replanted into a larger pot. When indoor plants such as those in the dracaena family, get root bound, the roots are unable to take up water, and the plant starts to droop. This stresses the plant, the edges of the leaves begin to turn brown from a lack of water, and they become susceptible to other pathogens (fungi, bacteria, some viruses), which causes the spots you see.  
Since the spots appear to be irregular in shape, I suspect they are fungal in nature, often accelerated by high humidity in the house. They turn yellow and then become necrotic (dead, blackened tissue) in the middle, and unsightly. Washing off the plant's leaves may also help remove fungi before they infect the plant.

Here is an article on houseplant care, including instructions for repotting it into a larger, draining container:  https://extension2.missouri.edu/g6510

Hope this is helpful.  Good luck!

Thank you so much. I love these plants. I really want to save them. I will repot them tomorrow. After washing the leaves, can I use a spray to assist in killing the fungi? I truly appreciate your help. 

Oh, one more thing...should I cut the leaves (or the parts of the leaves) that are infected off?

You’re welcome! What we don’t know is whether there are still fungal spores, or whether they are no longer active. You don’t want to use a fungicide if it’s not needed. It does not work preventatively, although there is evidence sulfur might. So pass on that, and see what happens after repotting. Let me know what’s happening!

Since we don’t know whether the fungus ia still active or not, pruning won’t help it look better, but may abate the infection.