Sheriff (unbuttoning his overcoat and stepping away from the stove as if to the beginning of official business).

how does the use of the word “official” in this stage direction affect the mood of the scene?

Official is being used as it is in a serious mind set.

The mood of the scene refers to the feeling that a scene transmits to the viewer. In this scene, we learn about a sheriff and his behaviour. We learn that the sheriff is professional and responsible. He has mannerisms that show when he is about to conduct official business. The use of the word "official" gives the scene a serious mood.

appeal to ethos is your answer.