Your short quote refers to how Emily looked when the new leaders of the town went to visit Emily to try to collect her taxes.

Her skeleton was small and spare; perhaps that was why what would have been merely plumpness in another was obesity in her. She looked bloated, like a body long submerged in motionless water, and of that pallid hue. Her eyes, lost in the fatty ridges of her face, looked like two small pieces of coal pressed into a lump of dough as they moved from one face to another while the visitors stated their errand.

Like the Old South, Emily has aged, but not well. She seems to be dead already, barely clinging to a life that is no more. She's alone and sad. She's unaware that Col. Sartoris is dead, so she's lost track of time. She's literally lost in her own world, and she's barely hanging on to it. The condition of her house mirrors her physical appearance of being decayed and decrepit. Just like her house is falling down around her, her body and mind are also failing her. This description serves to foreshadow the gruesome discovery at the end of the story.