Is what the questioner is doing correct by ignoring his unveiled female cousin, speaking with her only when necessary, and not being in seclusion with her?

The questioner is thanked for his eagerness to maintain ties with his aunt while avoiding forbidden acts. He was correct in refraining from looking at, being in seclusion with, or speaking to his aunt's daughter, because she is not one of his mahram. Allah Almighty says: "and the daughters of your paternal uncles and the daughters of your paternal aunts" (Quran 33:50). It remains for him to alert his aunt's daughter that it is not permissible for her to sit before him without hijab. This alert should be conveyed through his aunt or mother, without direct communication.