Large language models (LLMs) are increasingly proposed as conversational agents in healthcare, yet many existing systems treat roles as static prompts and rely on one-shot safety filters. In such designs, it can be difficult to enforce long-horizon responsibilities, stable role identity, and realistic communication behavior. We propose a Self-Evolving LLM Agent that learns from role-based social experience and explicitly models communicator-level individual traits informed by prior communication questionnaires and clinical literature. The agent integrates (i) perception and action conditioned on both hard role responsibility norms and soft trait-conditioned style preferences, (ii) structured memory storing norm-annotated trajectories and identity states, (iii) dual-layer reflection that combines short-term responsibility diagnosis with long-term identity drift detection via trait consistency and trait-norm compatibility checks, and (iv) self-evolution that updates system prompts and identity parameters through preference-style optimization with AI feedback. We instantiate the framework in a multi-role healthcare sandbox and evaluate outpatient medication review, emergency triage, and discharge planning. Across our simulated tasks, self-evolution is associated with lower severity-weighted norm risk, more stable role-identity signals, and improved social embeddedness metrics (including trust-like signals) relative to strong static baselines.