This paper introduces a full solution for decentralized routing in Low Earth Orbit satellite constellations based on continual Deep Reinforcement Learning (DRL). This requires addressing multiple challenges, including the partial knowledge at the satellites and their continuous movement, and the time-varying sources of uncertainty in the system, such as traffic, communication links, or communication buffers. We follow a multi-agent approach, where each satellite acts as an independent decision-making agent, while acquiring a limited knowledge of the environment based on the feedback received from the nearby agents. The solution is divided into two phases. First, an offline learning phase relies on decentralized decisions and a global Deep Neural Network (DNN) trained with global experiences. Then, the online phase with local, on-board, and pre-trained DNNs requires continual learning to evolve with the environment, which can be done in two different ways: (1) Model anticipation, where the predictable conditions of the constellation are exploited by each satellite sharing local model with the next satellite; and (2) Federated Learning (FL), where each agent's model is merged first at the cluster level and then aggregated in a global Parameter Server. The results show that, without high congestion, the proposed Multi-Agent DRL framework achieves the same E2E performance as a shortest-path solution, but the latter assumes intensive communication overhead for real-time network-wise knowledge of the system at a centralized node, whereas ours only requires limited feedback exchange among first neighbour satellites. Importantly, our solution adapts well to congestion conditions and exploits less loaded paths. Moreover, the divergence of models over time is easily tackled by the synergy between anticipation, applied in short-term alignment, and FL, utilized for long-term alignment.