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Dolphus wants people to think of him as a drunk and a miscreant to draw attention from the fact that his real "crime" as they might see it, is his relationship with a black woman.
This allows townspeople to excuse his black mistress, and they chalk it up to his drinking. Dolphus only uses the drinking as an excuse to circumvent the ignorance and bigotry of the town. Dolphus simply prefers the company of blacks. The people do not know he is only drinking Coke.