Is it permissible for us to sit without hijab or shake hands with those whom my father adopted, after we found out that they are not our siblings, or should they be treated as strangers?

If your father's wife breastfed these children a valid, Shariah-compliant breastfeeding, then they are considered your siblings through suckling. You are permitted to sit with them without hijab and shake their hands, based on the view of the جمهور (majority of scholars) who hold that "the milk of the male" (i.e., the husband's seminal contribution that leads to the wife's lactation) establishes mahramiyyah (prohibition of marriage), citing as evidence the hadith of Aisha (may Allah be pleased with her) regarding the story of Aflah, the brother of Abu al-Qu'ays.