Chinese text correction has traditionally focused on spelling and grammar, while factual error correction is usually treated separately. However, in paragraph-level Chinese professional writing, linguistic (word/grammar/punctuation) and factual errors frequently co-occur and interact, making unified correction both necessary and challenging. This paper introduces CLFEC (Chinese Linguistic & Factual Error Correction), a new task for joint linguistic and factual correction. We construct a mixed, multi-domain Chinese professional writing dataset spanning current affairs, finance, law, and medicine. We then conduct a systematic study of LLM-based correction paradigms, from prompting to retrieval-augmented generation (RAG) and agentic workflows. The analysis reveals practical challenges, including limited generalization of specialized correction models, the need for evidence grounding for factual repair, the difficulty of mixed-error paragraphs, and over-correction on clean inputs. Results further show that handling linguistic and factual Error within the same context outperform decoupled processes, and that agentic workflows can be effective with suitable backbone models. Overall, our dataset and empirical findings provide guidance for building reliable, fully automatic proofreading systems in industrial settings.