If we were contracting "I do not need to go" and "I do not want to go", we could indeed say "I do not need nor want to go."
Here, however, due to the nature of the verbs involved, the two statements we want to combine do not share the same construction: "I cannot go" and "I do not want to go". In negative statements, ordinary verbs require do-support (as in "do not want"), but this construction would not work with modal auxiliaries such as "can". Therefore, "I cannot nor want to go" feels awkward, because the second verb "want" is lacking the expected do-support. The somewhat less-elliptical "I cannot go, nor do I want to", on the other hand, fulfills this expectation and therefore feels proper.