You are right to say that fat can be used as a noun or as an adjective. This is illustrated in the following sentences:
There’s a lot of fat on this meat. (Here, fat is a noun).
She’s got fat since she stopped exercising. (Here, fat is an adjective).
However, I think the object of the exercise is to highlight the attitude of the speaker. Fat and overweight can be used to describe the same person; the speaker shows his attitude through his / her choice of adjective. Fat would be considered pejorative whereas overweight is slightly more complimentary to the person. So, a doctor is more likely to tell somebody that they are overweight; (s)he probably wouldn’t tell the patient that (s)he is fat. If you want to be pejorative, you would probably say “she’s fat and ugly”; not “she’s overweight and ugly”.
I hope this will help you and good luck for the rest of your course.