Get your trainer to longe you, so you acquire a really independent seat, riding without reins and stirrups and doing exercises on the longe line.
That and riding outside on all kinds of terrain will get your skills and balance up to any kind of horse.
As far as the horse leaping into the canter, that takes training with many transitions to even out.
Coming from a western background, your trainer should be more than familiar with how to address that in a horse, any horse, as smooth transitions from any gait is very important in the western disciplines that are more technical, like reining and such.
Don’t despair, it will come for both of you, your own balance and the horse’s both.
I remember when I first started as an assistant instructor I had one OTTB that was very bad, took several weeks to get that horse to come back to hand and not be all strung out.
I so wanted the main trainer to get on and get the horse performing well, as he would have, but he said no, I had to learn to get there on my own.
It was so easy once we both could do smooth transitions without hurrying into them and it was a skill that stayed with me for all horses afterwards.
Keep working at this, but don’t work too much or too long and keep experimenting, don’t keep doing the same time and again if it is not working, said someone that spent too long on repetitive circles, before learning that changes is what does it, not trying to find balance first on it’s own.
Cavaletti, broom polo, hills, with good lessons on the longe line for both of you to keep any bad habits in check, that will get you there.
Now, your horse may always have lofty gaits, compared with western horse’s way of going, but if you like your horse, that is what he is.
I know some very strung out OTTBs years ago that, once on cattle, learned to use themselves so well, they didn’t even look like a TB any more when working, until they stood there or moved out.