You’ve gotten such good comments here, I am only going to mention one thing from my own experience.
Whatever the reasons, “your” horse (the horse you are riding) is clearly so inconsistent about maintaining a forward travel, it is bobbling both of you back and forth. I’m guessing it’s hard to keep his attention on top of everything else, and he’s reacting to everything without much regard for the rider.
Your body can never, ever get ahead of his motion - and that’s hard. It’s all in how you sit and in your awareness. If he hesitates, you don’t tip forward. If he dodges a bit right or left, you don’t tip forward.
The others who ride better than I do, teach more and know more, may hate this, but as a fellow ammy, you have to adopt a slightly westernized seat to keep your balance on a horse that is always dropping back. Your shoulders have to stay so far back, always behind his center of motion, you’ll feel you are leaning back in a lounge chair (and everyone will say “oh you sit up so straight!” :lol:). Below your hips, you need a chair seat with your feet somewhat ahead of the center of balance, legs long and heels down. No matter what he does, if you maintain this position then it can’t put you forward - so long as you are always paying attention and are never flustered by his behavior. You have to think calm, and be calm.
If you ‘perch’ in an equitation-style position up over his center of gravity and maybe even his shoulders, I think it’s not going to be workable with this horse, he’ll always be putting you on a bad balance. And he knows that already.
Whatever dressage trainers tell you about their ideal seat, if you carefully watch successful grand prix riders on highly reactive/sensitive horses, you’ll see some rider feet held a bit forward and a smidge of chair-seat, while the horse moves so beautifully through the movements. :winkgrin:
As a very good trainer said to me about one particular horse, “You can ride this horse, but you’ll have to ride like a pro every instant.” Your horse sounds like that’s the ride he needs. Given what you describe OP, particularly since this not even your horse, I’m not sure I’d be interested in making that investment of my energy and time. Whatever I might prove to myself by sticking it out and getting the horse going better is likely to be undone by the next rider. That’s could be a major reason why this horse is like this - inconsistent riding and a sensitive horse.
Will be interested in what you decide to try and how it all works out. Good luck finding a great riding experience, here or elsewhere!