“Busy” means having a lot to do or being engaged in many activities. Some synonyms for “busy” are “occupied”, “engaged”, and “hectic”.
“Occupied” implies being preoccupied or having one's attention taken up. For example, “She was occupied with work all day.” This shows that she was fully engaged in her work and had no free time.
“Engaged” can also mean being involved or committed to an activity. “He is engaged in a project that requires a lot of time and effort.” Here, “engaged” indicates that he is deeply involved in the project.
“Hectic” suggests a state of frenzied activity or chaos. “The office was hectic during the busy season.” This shows that there was a lot of commotion and activity going on.
These synonyms can be used to add variety and nuance to your language when describing a busy state or person.