> After a few seconds of "thinking", I eventually give a response that is no different to my knee-jerk reaction.
In interviews with Magnus Carlsen and the other top players, they all seem to say the same thing: the big difference between a short game (bullet/blitz) and a long game is that they'll have more time to verify the move in the long game. They don't spend more time finding the move.
I find this resonates with me. Very often the instinctive solution I come up with on the spot is often a very good solution and requires only minor tweaks.
> I internally start to panic (mainly thinking whether I have considered everything and the response is correct)
I just add a caveat: "I think X is a good solution but I have not had time to consider all the edge cases, so I will have to verify and come back to you" or similar.
Or, if the problem is complicated I'll just say that: "There are a lot of complexities/edge cases to consider here, I need to think more thoroughly about this. I'll get back to you later".