In George Washington Cable's "Tite Poulette" and Grace King's "The Little Convent Girl," the authors address the precarious position of the quadroon female. Although the outcomes of these narratives are quite different, there are many parallels between their fates. How can I write an essay analyzing the similarities and differences between Tite Poulette and the little convent girl and incorporate literary terms in my answer?

It sounds as though you are struggling to figure out an approach to this fairly complex assignment. Therefore, this answer will focus on helping you organize your paper into section and discuss the major points on which you should focus in each section.
Since the assignment specifies that you should not use outside resources, you should base your theoretical approach on material from your class notes. If you have been studying postcolonial theory, two important concepts your instructor may have discussed are "hybridity" and "passing". If so, the introduction to your paper should point out that both stories can be approached from these perspectives as they are both about young women who appear to be of mixed white and black blood, although we do discover that Tite eventually is shown to be of Spanish rather than African-American descent.
Your second section should give some historical background on Louisiana in the nineteenth century, and in particular its cultural hybridity. This should include not only the mixture of French, Spanish, and African-American cultures but also the practice of plaçage (a practice somewhere between concubinage and contract marriage). It would also be worth discussing the mythologizing of quadroon and octoroon women as sexually alluring but ultimately tragic figures.
Your third section should describe how passing is treated in both stories. Tite is a white girl being "passed" as part black so that she can stay with her mother, but she needs to assert her white identity to marry her beloved. The Little Convent Girl is of mixed heritage, capable of passing in white society (by, inter alia, obsessively straightening her hair), but in the world outside the convent there is no place for her, and thus she chooses to end her life rather than live it as a mixed-race woman.
The final section of the paper might discuss the social constructedness of race. Tite, for example, is alternately described as white and black, accepted as an octoroon until it turns out that she is Spanish. The Little Convent Girl is treated as white (despite some hints about her physical appearance) until her black mother appears in the story. This suggests that "race" is not handled by the authors as something genetic but rather as a matter of social role.