The iterative search process of evolutionary algorithms (EAs) encapsulates optimization knowledge within historical populations and fitness evaluations. Effective utilization of this knowledge is crucial for facilitating knowledge transfer and online adaptation. However, current research typically addresses these goals in isolation and faces distinct limitations: evolutionary sequential transfer optimization often suffers from incomplete utilization of prior knowledge, while adaptive strategies, utilizing real-time knowledge, are limited to tailoring specific evolutionary operators. To simultaneously achieve these two capabilities, we introduce the Optimization Knowledge Adaptation Evolutionary Model (OKAEM), a unified learnable evolutionary framework capable of adaptively updating parameters based on available optimization knowledge. By parameterizing evolutionary operators via attention mechanisms, OKAEM enables learnable update rules that facilitate the utilization of optimization knowledge via two phases: pre-training to integrate extensive prior knowledge for efficient transfer, and adaptive optimization to dynamically update parameters based on real-time knowledge. Experimental results confirm that OKAEM significantly outperforms state-of-the-art sequential transfer methods across 12 transfer scenarios via pre-training, and surpasses advanced learnable EAs solely through its self-tuning mechanism in prior-free settings. Beyond demonstrating practical utility in prompt tuning for vision-language models, ablation studies validate the necessity of the learnable components, while visualization analyses reveal the model's capacity to autonomously discover interpretable evolutionary principles. The code can be accessed at https://gitee.com/Anonymity_Paper/code-of-okaem.