The co-location of multiple database instances on resource constrained edge nodes creates significant cache contention, where traditional schemes are inefficient and unstable under dynamic workloads. To address this, we present SAM(a Stability-Aware Manager), an autonomic cache manager that establishes decision stability as a first-class design principle. It achieves this through its core control policy, AURA(Autonomic Utility-balancing Resource Allocator), which resolves the classic exploitation-exploration dilemma by synthesizing two orthogonal factors: the H-factor, representing proven historical efficiency (exploitation), and the V-factor, for estimated marginal gain (exploration). Through this practical synthesis and adaptive control, SAM achieves sustained high performance with strategic stability and robustness in volatile conditions. Extensive experiments against 14 diverse baselines demonstrate SAM's superiority. It achieves top-tier throughput while being uniquely resilient to complex workload shifts and adversarial workloads like cache pollution. Furthermore, its decision latency is highly scalable, remaining nearly constant as the system grows to 120 databases. Crucially, SAM achieves superior decision stability -- maintaining consistent optimization directions despite noise, avoiding performance oscillations while ensuring predictable Quality of Service. These results prove that a principled, stability-aware design is essential for sustained high performance in real-world, large-scale systems.