I'll be blunt: you are coming across to me as a doormat. Doormats get stepped on and people wipe their feet on them. That superior of yours is stepping on you and is wiping the dirt off his feet on you.
You need to ask yourself if this superior's hurry is more legitimate than your other superiors' hurry? What is it about this superior's hurry that makes it so special, aside that the hurry is his? When was the last time that he asked you to do something, and he was not in a hurry? My best guess is that he is always in a hurry.
When I tell others that I need to check with other superiors, I mean it. If I need to check with other superiors, that's because I need to do it and it's part of my job to make sure that I do it properly. I am not just playing with words. You say it but you don't mean it. Because if he tried to bull his way through it, he should have run into a brick wall. Your story reminds me of the story of The Three Little Pigs The next time you tell him you need to check with your superiors, you need to stick to what you're saying. Otherwise, he'll continue thinking that your words count for nothing unless and until he extracts or extorts a "Yes!" from you. At which point, he'll hold you to that "Yes!" as if you had given it to him of your own free will.
And frankly,I am not that far from thinking that you gave that "Yes!" freely to him because you had the feasible option to make a stand and you did not make a stand - Refer to Mahatma Gandhi for the difference between making a stand and putting up a fight. Mahatma Gandhi made stand after stand without ever putting up a fight.
You say that you have a "fight or flight" response when confronted and cornered. I don't see either "fight" or "flight" from you. You are obviously not fighting, and you are obviously not fleeing because the whole purpose of his bullying is to corner you and hem you in so that you can't take flight. So what does a prey do when it won't or can't fight and it can't run? It freezes in place, and that's exactly what you are doing. He has you frozen in place and agreeing to anything he wants, and he'll continue doing it until you find in yourself the spine to tell him to get off the bus. And he'll continue doing it to you for the simplest of reasons: it works for him. It's up to you to make him stop. It doesn't matter how clumsily you make him stop as long as you make him stop.
You could go to higher management or to HR for assistance because he is obviously bullying you into compliance, but this is a situation where a strong subordinate would tell a manager where to get off. Actually, I would have an eyeball to eyeball confrontation with him, AND I would report him to his management and to HR. And if his management wants a conference with him, I will be more than happy to attend the conference and educate everyone as to his tactics. As US Supreme Court Justice Brandeis used to say, sunlight is the best disinfectant. By the time I am done with him, he'll be voiding himself into his pants the next time he so much as looks at me.