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While Large Language Models demonstrate immense potential as proactive Medical Agents, their real-world deployment is severely bottlenecked by data scarcity under privacy constraints. To overcome this, we propose State-Enhanced Logical-Skill Memory (SELSM), a training-free framework that distills simulated clinical trajectories into entity-agnostic operational rules within an abstract skill space. During inference, a Query-Anchored Two-Stage Retrieval mechanism dynamically fetches these entity-agnostic logical priors to guide the agent's step-by-step reasoning, effectively resolving the state polysemy problem. Evaluated on MedAgentBench -- the only authoritative high-fidelity virtual EHR sandbox benchmarked with real clinical data -- SELSM substantially elevates the zero-shot capabilities of locally deployable foundation models (30B--32B parameters). Notably, on the Qwen3-30B-A3B backbone, our framework completely eliminates task chain breakdowns to achieve a 100\% completion rate, boosting the overall success rate by an absolute 22.67\% and significantly outperforming existing memory-augmented baselines. This study demonstrates that equipping models with a dynamically updatable, state-enhanced cognitive scaffold is a privacy-preserving and computationally efficient pathway for local adaptation of AI agents to clinical information systems. While currently validated on FHIR-based EHR interactions as an initial step, the entity-agnostic design of SELSM provides a principled foundation toward broader clinical deployment.