You have to use something else. Maybe something like this https://www.amazon.com/Picture-Hangers-Shelves-Planters-Decorations/dp/B07XD8CQBN/ (example, not endorsement, do your research)

Caulk is not glue. Do not use glue, especially in a rental. Any glue that's strong enough to hold a nail in drywall will cause more damage when it comes time to remove the nail than if you had used a proper anchor in the first place.