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Generic text rewriting is a prevalent large language model (LLM) application that covers diverse real-world tasks, such as style transfer, fact correction, and email editing. These tasks vary in rewriting objectives (e.g., factual consistency vs. semantic preservation), making it challenging to develop a unified model that excels across all dimensions. Existing methods often specialize in either a single task or a specific objective, limiting their generalizability. In this work, we introduce a generic model proficient in factuality, stylistic, and conversational rewriting tasks. To simulate real-world user rewrite requests, we construct a conversational rewrite dataset, ChatRewrite, that presents ``natural''-sounding instructions, from raw emails using LLMs. Combined with other popular rewrite datasets, including LongFact for the factuality rewrite task and RewriteLM for the stylistic rewrite task, this forms a broad benchmark for training and evaluating generic rewrite models. To align with task-specific objectives, we propose Dr Genre, a Decoupled-reward learning framework for Generic rewriting, that utilizes objective-oriented reward models with a task-specific weighting. Evaluation shows that \approach delivers higher-quality rewrites across all targeted tasks, improving objectives including instruction following (agreement), internal consistency (coherence), and minimal unnecessary edits (conciseness).