The only way "Sono pieno" makes sense for "I'm busy" is when you are asking for an appointment, for example with a doctor, and you ask, "Do you have any openings on Tuesday?" Then "Sono pieno" makes sense, in the context of "I'm full that day." Otherwise, "I'm busy" would be "Sono impegnato/a" OR "Sono occupato/a." I really wish Duolingo would not offer unlikely alternative meanings without explaining how they are used.
Yes, it is full of food. But Duolingo gave an alternate meaning of "busy," which isn't quite right.
Thank you for your response. However, using an Italian-English dictionary, I find that "impregnato" means "impregnated" or "full of". However, the word "impegnato" can be translated as "busy." So I think there was a typo in your reply. Either way, sono pieno, seems to be properly translated as "I am full."
Unless you are at dinner in some areas of Northern Italy, where "sono piena" will be understood as saying "I'm pregnant" (see comment below). When in Florence more than 40 years ago, a female friend was informed of this by an elderly professor, and the scene has remained fresh in my mind ever since.