Large language model (LLM) agents are increasingly applied to network troubleshooting, but root-cause localization on public benchmarks remains well below practical deployment thresholds. We argue this is because existing agents do not encode the disciplined, layer-by-layer methodology that human network engineers use, and instead rely on free-form deliberation that conflates evidence acquisition with hypothesis commitment. We present SADE (Symptom-Aware Diagnostic Escalation), an agent that encodes the classical Cisco troubleshooting methodology as an explicit policy. SADE pairs a phase-gated diagnostic workflow, which separates evidence acquisition from hypothesis commitment, with a routed library of fault-family skills and high-yield diagnostic helpers. On a held-out 523 incident set of the public NIKA benchmark covering eleven unseen scenarios, SADE improves root-cause F1 by 37 percentage points over a ReAct + GPT-5 baseline; a model-controlled comparison against the same Claude Sonnet backend without the SADE policy attributes 22 of those points to the diagnostic policy alone, showing that the gain is not a side-effect of the model upgrade.