Prompt-based in-context learning (ICL) and parameter fine-tuning are two dominant paradigms for incorporating external information into large language models (LLMs), but they incur high inference costs or require expensive retraining. To bridge this gap, context-to-parameter mapping converts prompts into temporary adapter weights. However, we identify a critical failure mode in existing methods: *hidden-state collapse*, where the adapter-augmented model’s internal states diverge sharply from the full-context oracle in deeper layers. We trace this failure to two coupled gaps: suboptimal **Input-Selection** and inadequate **Supervision-Signal**. To address these issues, we propose SADA (**S**tate-**A**ligned **D**istillation **A**dapters). We establish the *attention-block output* as a principled feature interface to improve input selection and introduce *state-alignment distillation* to enforce consistency between the adapter-augmented model and the full-context oracle. Experiments on long-context language modeling (PG19) and downstream NLU and summarization benchmarks show that SADA consistently outperforms strong baselines like *StreamAdapter* and *GenerativeAdapter*, achieving performance comparable to ICL while significantly reducing memory footprint and latency. We further analyze when parameterized context compression is effective and when explicit context retention remains preferable. Our code is available at [https://github.com/Taylor-Gavel/SADA.git](https://github.com/Taylor-Gavel/SADA.git).