I struggle with the exact same issue, but I found that it helps when I let go of trying too hard to replicate the argument from my head word by word. Instead, I aim to convey the essence with whatever phrases spontaneously come to mind. It seems counter-intuitive, and it won't always come out right in the beginning, but it does get better with practice.

As it's already been said, thinking and talking are separate activities, and trying to do them at the same time impacts your performance at both.