Fine-grained service offloading in collaborative edge computing can realize full use of the limited resources of edge nodes to achieve efficient parallel computing. The existing research mainly focuses on service delay but pays insufficient attention to the network status, which will easily cause unbalanced resource utilization. Therefore, we propose a resource and delay aware fine-grained service offloading mechanism. First, we propose a novel network-adaptive service graph reconstruction algorithm to reduce the complexity of service offloading and the transmission delay, which includes service graph partition, dependency conflict detection and elimination, and service graph re-creation. Next, to better balance link and node resource utilization respectively, we propose original graph-based and association graph-based service graph mapping algorithms based on graph neural networks. A goal-directed affinity-based loss function is explored for them, which aims to address the difficulty of label generation in supervised learning. We conduct extensive simulation experiments with different numbers of subtasks, edge nodes and service requests under different network resource statuses. The experimental results show that the proposed service graph reconstruction method can balance network resource utilization, while reducing the service transmission delay and algorithm execution time for complex services. Moreover, the service graph mapping algorithms can improve the resource utilization balance degree while satisfying service constraints with start-end node location, resources and delay in various scenarios, especially in the case of unbalanced user distribution. Generally, our fine-grained service offloading mechanism enables short execution time and strong scalability, and is applicable to dynamic edge networks.