Another conflict of the story is Miss Emily's relationship with the people of the town, especially as time goes forward. She is isolated from the people of town because she holds herself to be "a high and mighty Gierson" which would have been the attitude she learned from her father. Back when she lived on the most important street in town and her father was alive, her name meant something! And even in the few years after her father's death is meant something -- it is why the town elders remitted her taxes, why the pharmacist didn't deny her demand for the poison, and why the men didn't demand to investigate the smell -- she held a status in the society poeple respected.
As time went on though, the attitudes of the townspeople changed. The old guard sent their children to her for china painting lessons, but the younger generation didn't. The old guard didn't demand any taxes, the younger generation came to house and challenged her about them. The older generation said is was disgrace that she was 'dating' Homer, but the younger generation felt a little sorry for her.
She "overcomes" these conflicts with her unflinching maintance of her status and position in society, even if no one else really sees it or believes it anymore. She sends away the men about the taxes and NEVER pays any attention to the yearly tax notice. She refuses to let them put postal numbers on her house. She never leaves the house at all. She doesn't reveal any of her secrets -- they only come out after her death.
The central conflict that Miss Emily faced in the story is the role her father played in her life and how he frightened away any suitors and thus prevented her from really living and allowing her to marry and live her own life. Note what is said about Miss Emily and her father:
None of the young men were quite good enough for Miss Emily and such. We had long thought of them as a tableau, Miss Emily a slender figure in white in the background, her father a spraddled silhouette in the foreground, his back to her and clutching a horsewhip, the two of them framed by the back-flung front door.
Later on the narrator mentions "We remembered all the young men her father had driven away". Clearly Miss Emily faces the central external conflict of gaining a husband to satisfy her desperate desire to love and be loved. This is why, after her father's death, she goes to such lengths to "secure" for herself a husband, even if that actually means securing him by killing him and lying next to his corpse for the rest of her life.