Cognitive and neurological impairments are very common, but only a small proportion of affected individuals are diagnosed and treated, partly because of the high costs associated with frequent screening. Detecting pre-illness stages and analyzing the progression of neurological disorders through effective and efficient intelligent systems can be beneficial for timely diagnosis and early intervention. We propose using Large Language Models to extract features from free dialogues to detect cognitive decline. These features comprise high-level reasoning content-independent features (such as comprehension, decreased awareness, increased distraction, and memory problems). Our solution comprises (i) preprocessing, (ii) feature engineering via Natural Language Processing techniques and prompt engineering, (iii) feature analysis and selection to optimize performance, and (iv) classification, supported by automatic explainability. We also explore how to improve Chatgpt's direct cognitive impairment prediction capabilities using the best features in our models. Evaluation metrics obtained endorse the effectiveness of a mixed approach combining feature extraction with Chatgpt and a specialized Machine Learning model to detect cognitive decline within free-form conversational dialogues with older adults. Ultimately, our work may facilitate the development of an inexpensive, non-invasive, and rapid means of detecting and explaining cognitive decline.