Emotional self-awareness is a crucial social-emotional skill for positive psychology and mental well-being. While prior works have used AI applications to foster emotion awareness, it is unclear what specific design features users prefer in the context of self-awareness skill practice. To better understand these features, we design, deploy, and evaluate Teddy, a mobile conversational agent that helps users practice verbalizing and reflecting on their emotions through experience sampling, affect labeling, and personal storytelling interactions. We conduct a user study with 18 days of interaction data from six participants to evaluate what aspects of Teddy contribute to emotion awareness. We find that users felt comfortable self-disclosing to the agent and found it helpful to verbalize their emotions, but desired more characteristics like social responsiveness and personalization. Based on our pilot study, we outline design insights for future work on AI agents that support practice of social-emotional skills such as emotional self-awareness.