The rapid expansion of cloud, fog, and edge services is pushing the Distributed Computing Continuum (DCC) toward unsustainable carbon footprints, while existing schedulers optimize latency or energy in isolation and therefore struggle to respect strict carbon budgets. We propose a carbon-aware task-scheduling framework that couples short-term performance with long-term sustainability. A Lyapunov-guided virtual queue converts horizon-wide emission constraints into per-slot stability targets, and a dynamically weighted Proximal Policy Optimization (PPO) agent allocates tasks in real time. After placement, a closed-form convex solver slices heterogeneous node resources, completing a dual-layer loop that blends theoretical guarantees with Deep Reinforcement Learning (DRL). Experiments on a 100-node DCC driven by carbon-intensity traces show that, under a tight 0.15 g CO \(_2\) eq budget, the proposed method attains 3.02 s average latency and 0.14g emissions, achieving 11% faster service and 7% lower emissions than the best-performing reinforcement learning baseline while reducing latency by 58% compared with uniform slicing. Dynamic weight adaptation stabilizes the virtual queue, whereas static-weight and no-convex variants either violate carbon limits or incur much higher delays. Consistent advantages across all tested budgets from 0.125 to 0.20g CO \(_2\) eq confirm the practicality of learning-driven, Lyapunov-constrained orchestration and open a path toward carbon-aware management of next-generation distributed infrastructures.