This study introduces a deep reinforcement learning (DRL) framework for hedging American put options, which pose unique challenges due to their early exercise feature and lack of closed-form pricing solutions. Leveraging the deep deterministic policy gradient (DDPG) method, we design agents capable of navigating continuous action spaces and adapting to complex market dynamics. A key contribution is a novel reward function tailored for American options, incorporating early exercise sensitivity and penalizing transaction costs quadratically to reflect market frictions. We conduct two rounds of experiments: the first uses simulated paths from a Geometric Brownian Motion (GBM) model, showing that the DRL agent outperforms traditional strategies such as Black-Scholes (BS) Delta hedging and binomial tree methods, especially under transaction costs and volatility misestimations. In the second, we train agents using market-calibrated stochastic volatility models, derived from 80 put options across 8 equity symbols. Using Chebyshev interpolation, we efficiently generate American option prices throughout training. The DRL agents not only achieve superior performance under simulated stochastic paths but also when tested against empirical asset prices from the actual market between option sale and maturity. This work presents the first DRL approach specifically calibrated for American-style options, and offers a scalable, model-agnostic framework for extension to more complex underlying processes.