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The advancements of remote sensing (RS) pose increasingly high demands on computation and transmission resources. Conventional ground-offloading techniques, which transmit large amounts of raw data to the ground, suffer from poor satellite-to-ground link quality. In addition, existing satellite-offloading techniques, which offload computational tasks to low earth orbit (LEO) satellites located within the visible range of RS satellites for processing, cannot leverage the full computing capability of the network because the computational resources of visible LEO satellites are limited. This situation is even worse in hotspot areas. In this paper, for efficient offloading via LEO satellite networks, we propose a novel computing-aware routing scheme. It fuses the transmission and computation processes and optimizes the overall delay of both. Specifically, we first model the LEO satellite network as a snapshot-free dynamic network, whose nodes and edges both have time-varying weights. By utilizing time-varying network parameters to characterize the network dynamics, the proposed method establishes a continuous-time model which scales well on large networks and improves the accuracy. Next, we propose a computing-aware routing scheme following the model. It processes tasks during the routing process instead of offloading raw data to ground stations, reducing the overall delay and avoiding network congestion consequently. Finally, we formulate the computing-aware routing problem in the dynamic network as a combination of multiple dynamic single source shortest path (DSSSP) problems and propose a genetic algorithm (GA) based method to approximate the results in a reasonable time. Simulation results show that the computing-aware routing scheme decreases the overall delay by up to 78.31% compared with offloading raw data to the ground to process.