Text sanitization aims to rewrite parts of a document to prevent disclosure of personal information. The central challenge of text sanitization is to strike a balance between privacy protection (avoiding the leakage of personal information) and utility preservation (retaining as much as possible of the document's original content). To this end, we introduce a novel text sanitization method based on generalizations, that is, broader but still informative terms that subsume the semantic content of the original text spans. The approach relies on the use of instruction-tuned large language models (LLMs) and is divided into two stages. Given a document including text spans expressing personally identifiable information (PII), the LLM is first applied to obtain truth-preserving replacement candidates for each text span and rank those according to their abstraction level. Those candidates are then evaluated for their ability to protect privacy by conducting inference attacks with the LLM. Finally, the system selects the most informative replacement candidate shown to be resistant to those attacks. This two-stage process produces replacements that effectively balance privacy and utility. We also present novel metrics to evaluate these two aspects without needing to manually annotate documents. Results on the Text Anonymization Benchmark show that the proposed approach, implemented with Mistral 7B Instruct, leads to enhanced utility, with only a marginal (< 1 p.p.) increase in re-identification risk compared to fully suppressing the original spans. Furthermore, our approach is shown to be more truth-preserving than existing methods such as Microsoft Presidio's synthetic replacements.