Mobile edge computing (MEC) enables computation-intensive and latency-sensitive tasks to be offloaded from mobile devices to nearby edge servers. Most existing MEC task offloading studies formulate offloading as a selection problem over a fixed or fully available set of candidate servers, which is restrictive in heterogeneous MEC scenarios with task-node eligibility constraints. Under such constraints, a task can be processed by an edge server only when task attributes, service requirements, link conditions, and node states jointly satisfy the corresponding eligibility conditions. The feasible action set therefore varies over time, while offloading decisions are further coupled with local queueing competition and long-term load evolution. To address this problem, this paper proposes RoSCo, a load-aware task offloading method with scheduling and constraint coordination for eligibility-constrained MEC systems. RoSCo constructs a dynamic feasible action set, applies eligibility-aware action masking to exclude infeasible offloading actions, introduces priority-driven local coordination to characterize service competition among heterogeneous tasks, and designs a load-responsive reward to guide congestion mitigation and load balancing. The offloading policy is learned using a dueling double deep Q-network (D3QN). Simulation results show that RoSCo reduces task drop rate and edge-node load imbalance while maintaining competitive task completion delay and energy consumption, especially under high-load and sparse-eligibility conditions.