Neural NLP models are often miscalibrated and overconfident, assigning high confidence to incorrect predictions and failing to express uncertainty during internal evidence aggregation. This undermines selective prediction and high-stakes deployment. Post-hoc calibration methods adjust output probabilities but leave internal computation unchanged, while ensemble and Bayesian approaches improve uncertainty at substantial training or storage cost. We propose UAT-LITE, an inference-time framework that makes self-attention uncertainty-aware via Monte Carlo dropout in pretrained transformer classifiers. Unlike output-level calibration (e.g., TS), UAT-LITE injects epistemic uncertainty directly into attention, enabling uncertainty-aware routing during contextualization and token-level diagnostic signals beyond global logit rescaling. Token-level epistemic uncertainty is estimated from stochastic forward passes and used to modulate self-attention during contextualization, without modifying pretrained weights or training objectives. We additionally introduce a layer-wise variance decomposition to diagnose how predictive uncertainty accumulates across transformer depth. Across SQuAD 2.0 answerability, MNLI, and SST-2, UAT-LITE achieves an average relative ECE reduction of approximately 20% compared with a fine-tuned BERT-base baseline while preserving accuracy, and yields more informative uncertainty behavior for selective prediction under distribution shift.