Am I liable for a right, or an expiatory offering for breaking an oath, or must I ask for forgiveness from a person who made me swear by Allah not to disclose a warning he gave me, but I disclosed it to my family and discussed parts of it with others?

Disclosing a secret is a betrayal of trust and forbidden if it leads to harm. You are required to offer expiation for breaking an oath due to disclosing the secret. This expiation is feeding ten poor persons, clothing them, or freeing a slave. If you cannot find any of these, then fasting for three days. You should seek forgiveness from the man, but if asking for his forgiveness will lead to greater discord, then supplicate for him and seek Allah's forgiveness for yourself, and do not inform him of what happened to avoid a greater corruption. This is what some scholars have chosen, because informing him is a corruption with no benefit and will cause him more harm.