Enabling large language models (LLMs) to utilize search tools offers a promising path to overcoming fundamental limitations such as knowledge cutoffs and hallucinations. Recent work has explored reinforcement learning (RL) for training search-augmented agents that interleave reasoning and retrieval before answering. These approaches usually rely on outcome-based rewards (e.g., exact match), implicitly assuming that optimizing for final answers will also yield effective intermediate search behaviors. Our analysis challenges this assumption: we uncover multiple systematic deficiencies in search that arise under outcome-only training and ultimately degrade final answer quality, including failure to invoke tools, invalid queries, and redundant searches. To address these shortcomings, we introduce DeSA (Decoupling Search-and-Answering), a simple two-stage training framework that explicitly separates search optimization from answer generation. In Stage 1, agents are trained to improve search effectiveness with retrieval recall-based rewards. In Stage 2, outcome rewards are employed to optimize final answer generation. Across seven QA benchmarks, DeSA-trained agents consistently improve search behaviors, delivering substantially higher search recall and answer accuracy than outcome-only baselines. Notably, DeSA outperforms single-stage training approaches that simultaneously optimize recall and outcome rewards, underscoring the necessity of explicitly decoupling the two objectives.