Participating respectfully in your father’s/family’s culture is not appropriation. You have a right to learn about your heritage and to learn about the cultures within your heritage.
Being mixed race or biracial should not hinder you reconnecting with your family’s culture. Preserving culture is more important than I can explain. My advice would be to research cultural appropriation; I say this because if you think participating in your own family’s culture is appropriation then you need to learn more about it. :) And to pursue learning about the Cherokee. The histories, the current events, the songs, the dances, the languages, the art, the crafts, etc. Pursue this respectfully, Participate when it is appropriate for you to do so. Ask questions. If you think you are doing something wrong, ask someone.