So I finished up the class a while ago and it was fun.
We always did a little improv game warm up then the sustained campaign with
Kids on Bikes reskinned to be D&Dish. It has a couple collaborative questions to establish the setting you are in which fit well with our improv group with a lot of those details being the center of our plot and adventuring focus.
About half way through the first game after some interaction someone asked me if I was a vampire and so I went with it instead of a warlock as I had originally planned. I talked with the DM afterwards and he was fine with it so instead of eldritch blast I would hypnotize people as a class appropriate power thing. I would also always make sure I was invited into places before entering and never eat or drink with the others.
About the third game the DM asked me "You're an elf right?" so I went with being an elf from then on.
The intro improv games leaned more and more into tying into our campaign, the improv game Five Things (where you get a topic and quickly rattle off five things for that topic) had topics like five names of dancers at The Lucky Halfling (Hefty Harold, Pudgy Paul, etc.), five treasures that might be at the bottom of the giant hole, and so on. One really neat one was to have two people pair off, the group assigns a random scene setting, and the two improv a short flashback scene between the two characters. This created stuff we later used for further interactions in the present and the DM tied them in as well.
The improv was easy and fun, I have since gone to a few short improv jam things where it was straight improv and I think I might do more.
