In British English, "exam results" is more common. In American English, "exam grades" or "exam scores" are preferred. Both work, but they have slightly different focuses: Exam scores usually refer to numerical results (e.g., 85 out of 100). Exam grades refer to letter grades (e.g., A, B, C). American English version: "Those students were very excited every time the exam grades (or exam scores) were announced."

I think that sentence is perfectly acceptable in American English. As for your other question, I would prefer “scores”, over “grades.” One word used in British English is “marks”, whereas “grades “ is more common in American English. Example: He received good marks when he was in school (British), vs. He received good grades when he was in school. (American).