Conversational search has become increasingly important in retrieval-augmented generation (RAG) systems, where users interact with AI assistants through multi-turn conversations containing context-dependent queries. We propose RCEM, a conversational dense retrieval model that distills the query reformulation capability of LLMs into the embedding model, enabling context-aware retrieval without explicit query rewriting during inference. Unlike prior conversational dense retrieval approaches that learn direct conversation-to-document matching, RCEM aligns conversational-query embeddings with rewritten-query embeddings, improving robustness under distributional shift. RCEM does not require conversational query-to-document relevance mappings for training, which are often expensive and difficult to obtain with high quality. Extensive experiments on QReCC, TopiOCQA, and TREC CAsT demonstrate that RCEM consistently outperforms strong conversational retrieval baselines, achieving particularly large gains under distributional shift, including up to 20% improvement in Recall@10. RCEM further extends the base embedding model with conversational query rewriting capability while preserving its original retrieval functionality, allowing both standalone and conversational queries to be encoded by a single model and searched against existing document indexes without rebuilding the retrieval database.