I believe the name of the short story by William Faulkner is "That Evening Sun."  Nine-year-old Quentin is the narrator of this story about Nancy.  She has several affairs with white men and becomes pregnant.

"The troubled race relations that have characterized the South throughout its history are the backdrop for ‘‘That Evening Sun,’’ even if they are not the main concern of the story." 

This story is about guilt, prejudice, and fear.  The story is very powerful and enlightening about the hatred and fear of mixed race children during that era as well as how the women who bore them are treated. 

This story has the theme for which you indicated you were looking, and was written by Faulkner, which you requested.  However, Nancy does not drown at the end of the story.  She does try to hang herself while she is in prison and she does give birth to a child whose heritage is in question.  Hope this helps.

The story that you are looking for is "Desiree's Baby," by author Kate Chopin whose main character's background and racial heritage is not known.

"Désirée is the adopted daughter of the Valmonde family. Madame and Monsieur Valmonde have raised Désirée since she was a toddler when they found her by the plantation's front gate. Despite the fact that her ancestry is unknown."

After Desiree marries a white plantation owner, she is very happy, they have a son.  As the baby grows he starts to exhibit African ancestry, since Desiree does not know her background, she believes it comes from her side.

 Armand says that clearly the child is not white and that therefore she is not white. Désirée denies this charge. In despair, Désirée writes to her mother for an explanation.

Of course Desiree's mother cannot give her daughter proof of her heritage, she was adopted from unknown origins. Desiree, who can't bear that her husband is so angry at her and will not love his son does not go to the home of her parents, but

"Instead of walking to the road that will take her home, Désirée walks across the fields and into the bayou (Swamp). She and her baby are never seen again." 

Go to the links below, the second one is a Kate Chopin site and offers a link to read the story online.

Good Luck