The word "engaged" does have both of your stated meanings in the dictionary. However, the connotation in American English of "engaged" without a qualifier after it (i.e. what the person is "engaged in") is usually "engaged to be married". In normal conversation, the word "busy" would be used instead; the first sentence of your example would more commonly be said, "are you busy with anything, Margaret?" or simply, "Are you busy, Margaret?"
You can use the word "engaged" to mean "busy" or "involved" in the context of a process or group: "He is engaged in getting the new healthcare bill repealed", means the majority of his time is consumed in this task or process. Similarly, "she's engaged with NOW" means that the "she" being referred to is a member of the National Organization of Women (a civil rights group). This second meaning is a little older and less common now, but is still seen.