As privacy issues are receiving increasing attention within the Natural Language Processing (NLP) community, numerous methods have been proposed to sanitize texts subject to differential privacy. However, the state-of-the-art text sanitization mechanisms based on a relaxed notion of metric local differential privacy (MLDP) do not apply to non-metric semantic similarity measures and cannot achieve good privacy-utility trade-offs. To address these limitations, we propose a novel Customized Text sanitization (CusText) mechanism based on the original 𝜖-differential privacy (DP) definition, which is compatible with any similarity measure.Moreover, CusText assigns each input token a customized output set to provide more advanced privacy protection at the token level.Extensive experiments on several benchmark datasets show that CusText achieves a better trade-off between privacy and utility than existing mechanisms.The code is available at https://github.com/sai4july/CusText.