Over half of US adults with Alzheimer disease and related dementias remain undiagnosed, and speech-based screening offers a scalable detection approach. We compared large language model adaptation strategies for dementia detection using the DementiaBank speech corpus, evaluating nine text-only models and three multimodal audio-text models on recordings from DementiaBank speech corpus. Adaptations included in-context learning with different demonstration selection policies, reasoning-augmented prompting, parameter-efficient fine-tuning, and multimodal integration. Results showed that class-centroid demonstrations achieved the highest in-context learning performance, reasoning improved smaller models, and token-level fine-tuning generally produced the best scores. Adding a classification head substantially improved underperforming models. Among multimodal models, fine-tuned audio-text systems performed well but did not surpass the top text-only models. These findings highlight that model adaptation strategies, including demonstration selection, reasoning design, and tuning method, critically influence speech-based dementia detection, and that properly adapted open-weight models can match or exceed commercial systems.