I tend to deal with strong feelings by giving myself over to the task in hand.
1. Hard work, keeping gas and air mask up for my wife. Drove home from (teaching) hospital, not at all aware of other traffic. Then (10.00ish am), after phone calls to our parents, cried myself to sleep with tears of joy and thanks.
2. Very different, memories much less clear: in a GP-run unit in a country town. Birth was induced because doctor had to get away. I recall feeling cross that the only person who seemed to know what she was doing was the sister in the unit.