A man marrying a paternal cousin who is raised with him
just being raised in the same house as cousins does not make you Mahrams (unmarriageable relatives) to her. It is, therefore, Haram (prohibited) for you to kiss her, shake her hand, or be in Khulwah (being alone with a member of the opposite sex) with her. You should make Tawbah (repentance to Allah) for all that you did out of ignorance.
If the situation is as you mentioned, it is permissible for you or any of your brothers to propose to her. The fact that you lived in the same house as her during childhood has no effect on engagement and marriage to her. May Allah grant us success. May peace and blessings be upon our Prophet Muhammad, his family, and Companions.