When we say that someone has "gotten busy with" something, we usually mean that they have become preoccupied or deeply involved in a particular task or activity. However, there are many other ways to express this idea using different words and phrases. For example, we could say that someone has "taken up" a new hobby or project, or that they are "immersed in" their work. Similarly, we might say that someone is "dedicating themselves to" a goal or that they are "focused on" a particular task. Other possible synonyms for "gotten busy with" might include phrases like "engrossed in," "embroiled in," "absorbed in," or "entangled in".