In conflict, people use emotional expressions to shape their counterparts' thoughts, feelings, and actions. This paper explores whether automatic text emotion recognition offers insight into this influence in the context of dispute resolution. Prior work has shown the promise of such methods in negotiations; however, disputes evoke stronger emotions and different social processes. We use a large corpus of buyer-seller dispute dialogues to investigate how emotional expressions shape subjective and objective outcomes. We further demonstrate that large-language models yield considerably greater explanatory power than previous methods for emotion intensity annotation and better match the decisions of human annotators. Findings support existing theoretical models for how emotional expressions contribute to conflict escalation and resolution and suggest that agent-based systems could be useful in managing disputes by recognizing and potentially mitigating emotional escalation.