Federated learning has emerged as a promising paradigm for collaborative model training while preserving data privacy. However, many existing FL methods implicitly assume that clients have sufficient computational and storage resources, making them less applicable in real-world scenarios with severe system heterogeneity. To address this, submodel extraction has recently gained attention as a promising strategy to tailor the global model to resource-constrained clients. Despite this progress, existing methods often suffer from noticeable performance gaps across clients and structural inconsistency in the extracted models, leading to degraded global performance and increased communication overhead. In this work, we propose FedLAGC, a novel federated framework that jointly tackles performance imbalance and communication inefficiency through Layer-Adaptive submodel extraction and Gradient Correction. Specifically, FedLAGC constructs client-specific submodels by selecting structurally important parameters according to layer-wise importance scores, ensuring both resource adaptiveness and architectural consistency. Additionally, we propose a lightweight correction mechanism that captures historical optimization drift, helping to align local updates with the global direction and reduce redundant communication. The rigorous convergence analysis of FedLAGC for system-heterogeneous federated learning under non-convex objectives is given. Extensive experiments on CIFAR-10 and CIFAR-100 with ResNet-18 and ResNet-34 under various system and data heterogeneity settings demonstrate the significant superiority of FedLAGC (up to 24\% accuracy improvement and 3.66$\times$ communication efficiency) over state-of-the-art methods.