This study addresses the optimal execution of large stock sell programs by introducing TT-DAC-PS (Twin-Target Deterministic Actor-Critic with Policy Smoothing), a deterministic actor-critic architecture that combines twin exponential-moving-average critic targets with pessimistic min backup, TD3-style target policy smoothing noise, delayed actor updates, and conservative Q regularisation to curb overestimation. Exploration uses Ornstein-Uhlenbeck (OU) noise with a hybrid schedule: deterministic episode-wise decay, variance-guided adjustment based on recent reward dispersion, and a Soft Actor-Critic (SAC)-style temperature that is learned and mapped to the noise scale. The environment integrates Almgren-Chriss (AC) trade impact with Limit Order Book (LOB) prices and volumes, normalised state features, per-step volume participation caps, and a utility-based reward. The trade execution algorithm is applied to LOB data for ten U.S. stocks. Performance is assessed against reinforcement-learning baseline algorithms, including Proximal Policy Optimisation (PPO), Soft Actor-Critic (SAC), and Advantage Actor-Critic (A2C), as well as alternative trade execution algorithms, including Time-Weighted Average Price (TWAP), Volume-Weighted Average Price (VWAP), and AC. The proposed model consistently reduces mean implementation shortfall percentage with competitive variance, outperforming classical baselines and standard reinforcement-learning benchmark models.