What is Paul searching for in "Paul's Case"?
One thing that Willa Cather makes clear, from the beginning until the end of the story, is that Paul is different. He is an elemental; he would rather exist in an environment, than be a full part of it. He witnesses life from a distance: At the Carnegie, he looks at the actors and is pleased to be in their environment. At the Cordelia St. gatherings, he observes without interacting. Even when in New York, Paul tends to keep to himself and just absorbs the dynamics of the people in the dining room.
This is because Paul cannot find a true niche in life. He tries to stand out in school by telling amazing stories which he cannot verify later. He dreads life in Cordelia; he also dreads school, and dreads his own home. It is no wonder that he realizes, in part 2, that this sense of complete dissatisfaction as been with him forever:
... he could not remember the time when he had not been dreading something. Even when he was a little boy, it was always there—behind him, or before, or on either side [...] the dark place into which he dared not look, but from which something seemed always to be watching him—and Paul had done things that were not pretty to watch, he knew...
The sense of dread that Paul lives in indicates that he will never be able to find that niche he seems to be looking for. Or is he, really? He knew that his end would come regardless; that he would never go to Cordelia Street again. He would much rather die than go back to a life where he does not belong. This is indicative even more than Paul knew that there was no niche for him in the first place. He seeks, and cannot find, a place to belong.