We introduce Target-Event-Agent Networks (TEA Nets) as a computational framework to extract subjects (``Agents"), verbs (``Events"), and objects (``Targets") from texts. Grounded in cognitive network science and artificial intelligence, TEA Nets are implemented as an open-source Python library. We test TEA Nets in three case studies, demonstrating the framework's ability to perform interpretable emotion detection, semantic frame analyses, and linguistic inquiries across conspiracy texts and textual responses generated by LLMs. In the LOCO conspiracy corpus, TEA Nets revealed that highly conspiratorial narratives (4,227 texts) linked personal pronouns (``I", ``you", ``we") with the same actions twice as frequently as low-similarity conspiracy narratives. High-conspiracy narratives connected person-focused elements (``you", ``people") through actions eliciting anger above the random baseline ($z = 2.63, p < .05$), a trend absent in low-similarity conspiracy narratives, which emphasized scientific actors (``researcher", ``scientist"). In the HOPE and CounseLLMe datasets of 212 (human) and 200 (LLM-based) psychotherapy transcripts, respectively, TEA Nets highlighted emotional differences. When expressing feelings, Claude 3 Haiku, GPT-3.5, and humans used sad words with higher frequency than random expectations but Haiku expressed sadness with lower emotional intensity than humans ($U = 1243.5, p = .036$). We discuss these differences in the context of psychotherapy training on LLM-simulated patients. Our results show that Target-Event-Agent Networks can extract relevant emotional, syntactic, and semantic insights from narratives, opening new avenues for text analysis with cognitive network science.