Recent progress in large language models (LLMs) has enabled them to communicate their confidence in natural language, improving transparency and reliability. However, this expressiveness is often accompanied by systematic overconfidence, whose underlying causes remain poorly understood. In this work, we analyze the dynamics of verbalized confidence estimation and identify answer-independence -- the failure to condition confidence on the model's own answer -- as a primary driver of this behavior. To address this, we introduce ADVICE (Answer-Dependent Verbalized Confidence Estimation), a fine-tuning framework that promotes answer-grounded confidence estimation. Extensive experiments show that ADVICE substantially improves confidence calibration, while exhibiting strong generalization to unseen settings without degrading task performance. We further demonstrate that these gains stem from enhanced answer dependence, shedding light on the origins of overconfidence and enabling trustworthy confidence verbalization.