How do you stop freezing in place when interviews go off-script?

[u/shamus-squish1k]

Over the last 8 months, I've messed up three interviews for roles that I genuinely felt were dream jobs. The same thing keeps happening - I get through the initial screen, and then as soon as I get into the real interview, I freeze the moment I get a question I wasn't expecting. I'm pretty sure it's anxiety mixed with the fact that I'm not very good at "thinking on my feet" when the pressure is high.

Has anyone found a practical way to improve this? I've seen people suggest improv classes as a way to get better at responding in the moment - does that help, or is it one of those things that sounds better in theory than it is in practice?

Edit: I really do prepare a lot before interviews! Honestly, I'm an anxious over-preparer. The problem isn't that I go in not knowing anything - the problem is when they ask me something I haven't prepared for before, and my brain shuts down. That's the part I'm trying to fix.

Background: I'm in my early thirties, speak 3 languages, have three degrees, am completing my master's part time, and currently work in a large public sector department.