I'm wondering, for adjectives such as 太い and 悲しい that also have a progressive "to be" verb counterpart (i.e., 太っている and 悲しんでいる), what is the difference between using the i-adjective form and the verb form?
E.g.: 太っている人 vs 太い人。 悲しんでいる人 vs 悲しい人。
First off, you cannot really think of ている as "progressive verb" because it represents more than that. Second, you're right that there are adjectives, but these ている forms are not forms of the adjective but rather forms of their respective verbs, 太る and 悲しむ.
ている in the case of 太っている represents a resultant state. 太る on its own means simply "to get fat," so if you say 私は太ります it doesn't mean "I am fat" but rather "I will get fat." So by saying 太っている you are expressing a resultant state, namely that you have gotten fat (太って) and still are fat (いる).
悲しむ, as it appears, means to be sad about something, and in the same way as the above, the resultant state refers to the state of being sad.
Now for meat'n'taters of what you're asking: the difference in usage. 太い refers generally to cylindrical objects, like a tube or a string or a tree or something with a large diameter. You can refer to a person as 太い but it would sound a little more.. mean, I think. For example, you could safely someone's fingers are 太い, supposing they weren't self conscious about it.
悲しい人 as it stands is a little strange, since when referring to other people we tend to say 悲しがっている (to "show signs" of being sad) or 悲しそう (look sad). More broadly, though, I don't think there is too much difference between 悲しむ and 悲しい. One is an adjective and one is a verb, and I think it's just made a little trickier because of the fact that they look similar. If you now the difference between うれしい and 喜ぶ I think it's the same sort of thing. Also when you ue 悲しむ you can use it with an object, so for example you can say 父の死を悲しんでいます. You can also say 父の死で悲しい, but note how the focus shifts. In English it might be the difference between "I am grieving my father's death" and "I am sad about my father's death." I think using the verb shows a feeling that's a little more pervasive and perhaps stronger.
Last, 悲しい can be used to describe things as well as people. So you can have a movie that is sad, a 悲しい映画, but you cannot have a 悲しんでいる映画.