"저는 먹으면서 노래합니다."
Translation:I sing while I eat.
I think the problem with your sentence is that "...I am eating" is on Present Continuous and the sentence in Korean is on Simple Present. The thing is the word "while" in English connects better with a continuous tense instead of a simple one. Changing "while" to "when", which is acceptable according to the Tips of this unit, makes it better/easier to understand: "I sing when I eat".