Unmanned Aerial Vehicles (UAVs) in disaster response require complex, queryable intelligence that onboard CNNs cannot provide. While Vision-Language Models (VLMs) offer this semantic reasoning, their high resource demands make on-device deployment infeasible, and naive cloud offloading fails under the low-bandwidth, unstable networks endemic to disaster zones. We present AVERY, an intent-driven adaptive split computing framework for efficient VLM deployment on resource-constrained platforms. AVERY is motivated by the observation that operator intent must be treated as a first-class system objective, since missions such as broad situational monitoring and precise, spatially grounded investigation require different semantic products, latency targets, and resource allocations. To reflect this, AVERY advances split computing beyond traditional depth-wise partitioning through a functional, cognitive-inspired dual-stream split: a high-frequency, low-resolution Context stream for real-time awareness, and a low-frequency, high-fidelity Insight stream for deep analysis. This design enables a hierarchical split strategy: computation is first separated by function, then partitioned depth-wise across edge and cloud when the Insight stream is required. A lightweight, self-aware onboard controller monitors network conditions and operator intent to select from pre-trained compression models, navigating the accuracy-throughput trade-off at runtime. Evaluated using LISA-7B in an edge-cloud setting under fluctuating network conditions, AVERY achieves 11.2% higher accuracy than raw image compression, 93.98% lower energy consumption than full-edge execution, and average accuracy within 0.75% of the static High-Accuracy baseline during dynamic adaptation. Overall, AVERY enhances mission efficiency and enables real-time, queryable intelligence in dynamic disaster environments.