Quantum Federated Learning (QFL) offers a promising framework to train quantum models across distributed clients while keeping data strictly local. Due to its simplicity and low communication overhead, Federated Averaging (FedAvg) is the standard aggregation choice in QFL literature. However, deploying QFL on practical hardware exposes a severe double-drift phenomenon: the global model is simultaneously derailed by client drift from non-IID data and hardware bias from noisy quantum gradient estimates. In this work, we first analyze the convergence of FedAvg under these realistic conditions, mathematically demonstrating that quantum hardware bias creates a persistent error floor that standard averaging cannot correct. To overcome this limitation, we propose Q-ANCHOR, a quantum-aware federated aggregation architecture that anchors server updates with zero-noise extrapolation while applying stateful client correction to suppress both client drift and hardware-induced bias. Our convergence theory proves that Q-ANCHOR successfully mitigates classical client drift while actively reducing the hardware-bias floor. Experimental results demonstrate that Q-ANCHOR achieves significantly more stable training than conventional FL baselines.