In an edge--cloud collaborative edge-computing environment, an edge node (EN) must decide whether each user task should be executed locally, forwarded to a remote service (or cloud) node (SN), or rejected. The EN observes its local state directly but receives the SN state only through an intermittently refreshed cache. Status updating and task control therefore form an asynchronous closed loop under partial observability. Freshness-driven schemes, including those based on Age of Information (AoI), do not directly value an update by its effect on subsequent task decisions. We propose CoSMO (Co-design of Semantic-state Management and Offloading), a cooperative event-driven reinforcement learning (RL) framework that coordinates semantic status management and selective offloading through realized task utility. CoSMO learns a compact representation of the heterogeneous SN service state. At the SN, a recurrent semi-Markov double deep Q-network (Double DQN) agent jointly selects send/no-send and the next decision interval. At the EN, a task-terminal off-policy value-learning agent makes hierarchical gate--route decisions from local observations and stale remote semantics. The agents maintain separate observations and value targets but share the same realized task-utility stream, without centralized execution. Across the evaluated workload families, CoSMO's reported relative improvement in on-time completion rate over the best-performing competing method averages 18.6%--21.2%. For capacity-aware decision accuracy across the three strict-overload points, the corresponding reported gains average 17.6%--$17.9%.

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