The demand for higher data rates and energy efficiency in wireless communication systems drives power amplifiers (PAs) into nonlinear operation, causing signal distortions that hinder performance. Digital Pre-Distortion (DPD) addresses these distortions, but existing systems face challenges with complexity, adaptability, and resource limitations. This paper introduces DRL-DPD, a Deep Reinforcement Learning-based solution for DPD that aims to reduce computational burden, improve adaptation to dynamic environments, and minimize resource consumption. To ensure safety and regulatory compliance, we integrate an ad-hoc Safe Reinforcement Learning algorithm, CRE-DDPG (Cautious-Recoverable-Exploration Deep Deterministic Policy Gradient), which prevents ACLR measurements from falling below safety thresholds. Simulations and hardware experiments demonstrate the potential of DRL-DPD with CRE-DDPG to surpass current DPD limitations in both local and remote configurations, paving the way for more efficient communication systems, especially in the context of 5G and beyond.