The advent of the 6G era aims for ubiquitous connectivity, with the integration of non-terrestrial networks (NTN) offering extensive coverage and enhanced capacity. As manufacturing advances and user demands evolve, space-air-ground integrated networks (SAGIN) with computational capabilities emerge as a viable solution for services requiring low latency and high computational power. Resource management within joint communication and computing-embedded SAGIN (JCC-SAGIN) presents greater complexity than traditional terrestrial networks. This complexity arises from the spatiotemporal dynamics of network topology and service demand, the interdependency of large-scale resource variables, and intricate tradeoffs among various performance metrics. Thus, a thorough examination of resource management strategies in JCC-SAGIN is crucial, emphasizing the role of non-terrestrial platforms with processing capabilities in 6G. This paper begins by reviewing the architecture, enabling technologies, and applications in JCC-SAGIN. Then, we offer a detailed overview of resource management modeling and optimization methods, encompassing both traditional optimization approaches and learning-based intelligent decision-making frameworks. Finally, we outline the prospective research directions in JCC-SAGIN.