Modern edge-cloud environments host highly distributed and heterogeneous computing resources whose availability fluctuates rapidly due to mobility, dynamic workloads, and elastic scaling. Services are often replicated across multiple computing nodes to balance the computational load among different sites. Traditional routing mechanisms, which are agnostic to computational load, are unable to ensure efficient service selection or fair workload distribution, resulting in bottlenecks, service degradation, and suboptimal utilization of edge resources. This work presents a centralized Computing-Aware Traffic Steering (CATS) framework that integrates runtime compute metrics into routing decisions, enabling global, fairness-driven traffic allocation across the edge-cloud continuum. The proposed system combines a Segment Routing over IPv6 (SRv6) data plane with a Multiprotocol BGP (MP-BGP) service overlay, while delegating optimization to a logically centralized Orchestrator. The orchestrator collects telemetry from Ingress and Egress Agents, respectively reporting traffic demands and computational capacities, via RESTful APIs, and computes per-Ingress routing weights through a min-max fairness optimization algorithm. These weights are enforced in the network using weighted ECMP, allowing optimal steering of service requests across available instances. The framework is validated on a fully containerized FRRouting-based testbed built with Containerlab. Experimental results show that the proposed framework adapts well to dynamic variations in both demand and capacity, achieving globally fair utilization of service instances. It also exhibits strong resilience, promptly reacting to service failures detected via the BGP Monitoring Protocol (BMP) and restoring balanced operation upon recovery. Overall, this work provides a carrier-grade, computing-aware orchestration framework that bridges network routing and compute resource management, paving the way for scalable future edge-cloud deployments.