I am not a grammar czar, but here goes...
I would not agree that while is "far more appropriate", unless you were trying to make a point about the relative timings of the two events. Hence, "I was playing Football yesterday, and my sister was cooking dinner" is perfectly normal and appropriate.
Your second version: " I was playing Football yesterday. And my sister was cooking dinner" is a bit more unusual. One does not routinely start a sentence with "And", unless one is making a specific comment: separating the two statements to achieve an effect. Reading this would make me think that there was something out of the ordinary about your sister cooking dinner: perhaps she resented doing it because she wanted to play football too?
By the way, one does not normally spell "football" with a capital "F".