I'm having a hard time understanding your description of what happened, but drunken walking? I might be picturing this wrong but droopiness doesn't sound like girthy ness to me. But again, I could be picturing this wrong.
Anyways, yes neoprene is the word, and it's a very comfy girth for a horse. It's also not uncommon for lesson horses to be girthy due to a multitude of reasons basically boiling down to they get used a lot by many different people. If the tack fits correctly and it's a behavioral issue, you probably aren't going to be able to fix it on a lesson horse yourself.
But for future reference and if you ever get a horse of your own that is girthy, loosening the girth when they display this behavior is basically rewarding it. Next time, they know that if they start wigging out in the cross ties when you begin to cinch the girth up, all will be solved because you loosen it and take that pressure off. I wouldn't worry about correcting the issue with the lesson horse, but just so you know :)
I'm glad that you stayed calm in that situation though, as that's the best thing to do in any situation with a horse! Great job!
We are not human beings having a spiritual experience. We are spiritual beings having a human experience. -Pierre Teilhard de Chardin