Post-Traumatic Stress Disorder (PTSD) remains underdiagnosed in clinical settings, presenting opportunities for automated detection to identify patients. This study evaluates natural language processing approaches for detecting PTSD from clinical interview transcripts. We compared general and mental health-specific transformer models (BERT/RoBERTa), embedding-based methods (SentenceBERT/LLaMA), and large language model prompting strategies (zero-shot/few-shot/chain-of-thought) using the DAIC-WOZ dataset. Domain-specific end-to-end models significantly outperformed general models (Mental-RoBERTa AUPRC=0.675+/-0.084 vs. RoBERTa-base 0.599+/-0.145). SentenceBERT embeddings with neural networks achieved the highest overall performance (AUPRC=0.758+/-0.128). Few-shot prompting using DSM-5 criteria yielded competitive results with two examples (AUPRC=0.737). Performance varied significantly across symptom severity and comorbidity status with depression, with higher accuracy for severe PTSD cases and patients with comorbid depression. Our findings highlight the potential of domain-adapted embeddings and LLMs for scalable screening while underscoring the need for improved detection of nuanced presentations and offering insights for developing clinically viable AI tools for PTSD assessment.