I don't see any problem with using the word "occupied", but I think "taken" or "filled up" would be more natural.

Yes, it is (provided the previous sentence established what it is earlier than or whether you're saying you couldn't schedule it as all available spots that were there were filled). You could include some more info and say: "I couldn't schedule the exam earlier than the 22nd of January as all the dates were occupied" or say "I couldn't schedule the exam until now as all the dates were occupied" depending on what you wanted to convey :)