Large language models (LLMs) have recently demonstrated success in decision-making tasks including planning, control, and prediction, but their tendency to hallucinate unsafe and undesired outputs poses risks. This unwanted behavior is further exacerbated in environments where sensors are noisy or unreliable. Characterizing the behavior of LLM planners to varied observations is necessary to proactively avoid failures in safety-critical scenarios. We specifically investigate the response of LLMs along two different perturbation dimensions. Like prior works, one dimension generates semantically similar prompts with varied phrasing by randomizing order of details, modifying access to few-shot examples, etc. Unique to our work, the second dimension simulates access to varied sensors and noise to mimic raw sensor or detection algorithm failures. An initial case study in which perturbations are manually applied show that both dimensions lead LLMs to hallucinate in a multi-agent driving environment. However, manually covering the entire perturbation space for several scenarios is infeasible. As such, we propose a novel method for efficiently searching the space of prompt perturbations using adaptive stress testing (AST) with Monte-Carlo tree search (MCTS). Our AST formulation enables discovery of scenarios, sensor configurations, and prompt phrasing that cause language models to act with high uncertainty or even crash. By generating MCTS prompt perturbation trees across diverse scenarios, we show through extensive experiments that offline analyses can be used to proactively understand potential failures that may arise at runtime. Code is available at https://sites.google.com/illinois.edu/astllm.