Nowadays, an important point that has attracted the attention of system planners is to maintain the adequacy of distribution networks under an acceptable range during interactions with multi-carrier energy systems under several uncertain sources. Achieving scalable and operational solutions to deal with the risk associated with these challenges has become a major concern among system planners. Thanks to the recent advances, hydrogen-based facilities offer distinctive opportunities to increase the flexibility and adequacy of distribution networks considering the high penetration of renewable energy sources (RESs). In this regard, the development of a comprehensive market-based mechanism by distribution network operators with respect to grid code considerations in the presence of hydrogen-based facilities is challenging. This study tries to fill the research gaps regarding the optimal coordination of the hydrogen-based energy hubs and distribution networks that host high RESs penetration. To this end, a multi-objective framework is developed to represent a risk-constrained operation problem for the distribution networks in the day-ahead scheduling process. The main objective of this study is 2-fold: (a) to preserve the adequacy of distribution networks by enhancing the self-sufficiency level and (b) to guarantee the flexible operation of distribution networks by mitigating the techno-economic and environmental risks. To attain a risk-averse strategy, a hybrid robust-stochastic programming model is used to handle the uncertain nature of renewable generation, demand consumption, and market price. Moreover, the proposed framework considers the load shifting program, as efficient ancillary service, to evaluate the impact of the deferrable loads on improving the performance of system operators' decision-making. The feasibility and robustness of the proposed mixed-integer linear programming strategy are investigated through a set of case studies on the IEEE 33-bus test system under the GAMS software environment. Numerical results demonstrate that if hydrogen-based energy hubs are employed by operator, the whole operation cost decreases by up to 77%, renewable power curtailment decreases by up to 92%, and CO₂ emission cost lessens by up to 50%.