Chinese text correction has traditionally focused on spelling and grammar, while factual error correction is usually treated separately. In paragraph-level Chinese professional writing, linguistic (word/grammar/punctuation) and factual errors can co-occur and interact. Moreover, editorial review leaves few observable errors in published professional texts, making controlled benchmark construction important. 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 transfer of a specialized corrector, 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 errors within the same context outperforms decoupled pipelines, and that agentic workflows can be effective with suitable backbone models. Overall, CLFEC provides a new benchmark for Chinese text correction research and practical guidance for proofreading systems. The dataset is publicly available at [https://github.com/jiu2021/CLFEC-Dataset](https://github.com/jiu2021/CLFEC-Dataset).