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The satellite-terrestrial networks possess the ability to transcend geographical constraints inherent in traditional communication networks, enabling global coverage and offering users ubiquitous computing power support, which is an important development direction of future communications. In this paper, we take into account a multi-scenario network model under the coverage of low earth orbit (LEO) satellite, which can provide computing resources to users in faraway areas to improve task processing efficiency. However, LEO satellites experience limitations in computing and communication resources and the channels are time-varying and complex, which makes the extraction of state information a daunting task. Therefore, we explore the dynamic resource management issue pertaining to joint computing, communication resource allocation and power control for multi-access edge computing (MEC). In order to tackle this formidable issue, we undertake the task of transforming the issue into a Markov decision process (MDP) problem and propose the self-attention based dynamic resource management (SABDRM) algorithm, which effectively extracts state information features to enhance the training process. Simulation results show that the proposed algorithm is capable of effectively reducing the long-term average delay and energy consumption of the tasks.