A sigh can signal relief, for sure, but actually when we have a lot of deep sighs while engaging with our survival physiology, that is more generally a sign that we are stirring up more than our system can deal with, and we are actually suppressing and managing that energy with the breath. This also would easily give someone a headache.

Next time, if you feel yourself going into a sigh, pay attention to what was happening just before. Stop the lesson. Get curious about what is happening. What is the emotional/sensational experience? There may be a lot there that the sighs are actually managing.