Community search aims to identify a refined set of nodes that are most relevant to a given query, supporting tasks ranging from fraud detection to recommendation. Unlike homophilic graphs, many real-world networks are heterophilic, where edges predominantly connect dissimilar nodes. Therefore, structural signals that once reflected smooth, low-frequency similarity now appear as sharp, high-frequency contrasts. However, both classical algorithms (e.g., k-core, k-truss) and recent ML-based models struggle to achieve effective community search on heterophilic graphs, where edge signs or semantics are generally unknown. Algorithm-based methods often return communities with mixed class labels, while GNNs, built on homophily, smooth away meaningful signals and blur community boundaries. Therefore, we propose Adaptive Community Search (AdaptCS), a lightweight framework featuring three key designs: (i) an AdaptCS Encoder that disentangles multi-hop and multi-frequency signals, enabling the model to capture both smooth (homophilic) and contrastive (heterophilic) relations; (ii) a memory-efficient low-rank optimization that removes the main computational bottleneck and ensures model scalability; and (iii) an Adaptive Community Score (ACS) that guides online search by balancing embedding similarity and topological relations. Extensive experiments on both heterophilic and homophilic benchmarks demonstrate that AdaptCS outperforms the best-performing baseline by an average of 11% in F1-score, retains robustness across heterophily levels, and achieves up to 2 orders of magnitude speedup over the strongest ML-based CS baselines.