We are engaged. May we go out to eat or for a walk together, just the two of us?

You may not be alone together (khalwat), because an engaged couple are still not mahram to each other.
An engagement is only a promise to marry and does not make lawful anything that is forbidden before the contract. The Prophet ﷺ said: "Let no man be alone with a woman except with her mahram." If you need to meet for marriage matters such as choosing a home, discussing finances or meeting a wedding planner, bring a mahram or family member, or meet in a busy public place without physical contact. In Malaysia, being alone together in a secluded place may also be the offence of khalwat under the state Syariah criminal Enactment, with no exemption for engaged couples. If the engagement period feels hard, consider bringing the contract forward.