Next-generation wireless networks will rely on mmWave/sub-THz spectrum and extremely large antenna arrays (ELAAs). This will push their operation into the near field where far-field beam management degrades and beam training becomes more costly and must be done more frequently. Because ELAA training and data transmission consume energy and training trades off with service time, we pose a cross-layer control problem that couples PHY-layer beam management with MAC-layer service under delay-sensitive traffic. The controller decides when to retrain and how aggressively to train (pilot count and sparsity) while allocating transmit power, explicitly balancing pilot overhead, data-phase rate, and energy to reduce the queueing delay of MAC-layer frames/packets to be transmitted. We model the problem as a partially observable Markov decision process and solve it with deep reinforcement learning. In simulations with a realistic near-field channel and varying mobility and traffic load, the learned policy outperforms strong 5G-NR--style baselines at a comparable energy: it achieves 85.5% higher throughput than DFT sweeping and reduces the overflow rate by 78%. These results indicate a practical path to overhead-aware, traffic-adaptive near-field beam management with implications for emerging low-latency, high-rate next-generation applications such as digital twin, spatial computing, and immersive communication.