Multi-Domain Multi-Task (MDMT) recommendation aims to provide personalized recommendations by leveraging information across multiple domains and tasks. However, existing methods often suffer from spurious correlations between irrelevant features and the target, leading to negative transfer. To address this, we propose a Stable and Adaptive Fusion (SAF) framework for MDMT recommendation. SAF introduces a weighted Hilbert-Schmidt Independence Criterion (HSIC) loss to decorrelate irrelevant features from the target, learning sample weights that promote stable (i.e., robust to spurious correlations) representations in both bottom and expert layers. We employ Random Fourier Features (RFF) to enable scalable computation of the HSIC loss. We further employ adaptive feature and expert gating to select these stable features, enabling the model to capture intricate cross-domain and cross-task dependencies. The learned sample weights are also used to reweight the MDMT loss during training. Experiments on large-scale datasets show that SAF outperforms state-of-the-art baselines by up to 2\% in AUC. To facilitate further research, we release a new industrial dataset with 30 million interactions across 3 domains and 2 tasks, with 300 features.