[Zanni]

Seconding "impromptu theater." Or, as I was going to say, improv comedy. Improv will make you much more comfortable with 1) trusting your instincts, 2) keeping your cool when you can't immediately think of what to say or get an unexpected reaction to what you do say.

If that doesn't sound like your jam, maybe look into a speech class or Toastmasters. I know you're already comfortable speaking in front of large groups, but you'll have an opportunity to practice extemporaneous speech (speech with very limited prep), which should be directly applicable to your situation.