Large language models (LLMs) often answer ambiguous inputs as if they had a single intended meaning. This is problematic because input ambiguity is an upstream source of uncertainty: when a prompt admits multiple valid interpretations, the key question is whether the model should answer directly at all. Existing input uncertainty methods often rely on LLM-generated statements or clarifications, but such transformations may already disambiguate the input before the ambiguity is measured. We propose a label-preserving perturbation framework for input ambiguity detection in which perturbations remain close to the original input while preserving its label. We instantiate this principle with two detectors: a black-box rule-based perturbation library and a white-box importance-guided method using model-internal signals. Experiments across four datasets in open-domain, conversational, and visual QA show that our methods outperform output-side uncertainty baselines and LLM-paraphrase-based perturbation baselines. Our results highlight the importance of detecting ambiguity before attempting to disambiguate it. Our code is available here.