Assuming that neither source data nor source model parameters are accessible, black-box domain adaptation (BBDA) represents a highly practical yet challenging setting, where transferable knowledge is limited to the predictions of a black-box source model. Existing approaches exploit such knowledge via pseudo-label refinement or by leveraging vision-language models (ViLs), but they often fail to reconcile the inherent discrepancy between task-specific knowledge from black-box models and language-aligned semantic priors of ViLs, resulting in suboptimal integration and degraded adaptation performance. To address this challenge, we propose adaptive Dual-Teacher Distillation with Subnetwork Rectification (DDSR), a framework that explicitly reconciles these complementary yet inconsistent knowledge sources. DDSR employs an adaptive prediction fusion strategy to integrate predictions from the black-box source model and a ViL, generating reliable pseudo-labels for the target domain. A subnetwork-based regularization mechanism mitigates overfitting to noisy supervision by enforcing output consistency and gradient divergency. Furthermore, progressively improved target predictions iteratively refine both pseudo-labels and ViL prompts, enhancing semantic alignment. Finally, class-wise prototypes are used to further optimize target predictions via self-training. Extensive experiments on multiple benchmark datasets demonstrate that DDSR consistently outperforms state-of-the-art methods, including those with access to source data or source model parameters.