The way I see it, first, you learn to ride along with the horse. With that in mind, the best thing you can do is ride a lot of horses, different types, until you are fairly comfortable with all of them. You learn a deep, independent seat while you develop the balance and muscles you need. You don't try to change the horse so much as go along with him while you jump, trail ride, ride patterns, gymkhana. You get the idea!
The next stage is the influencing of the horse, getting him on the bit and so forth, attention to his rhythm, his balance, his attentiveness. For this, I believe riding ONE horse is best, a horse who will not be ridden by a lot of different kinds of people. How else can you learn whether or not he is learning? It's at this point when a schoolmaster might be helpful, to learn the feel of what is correct. But just learning to "push the right buttons" has never been helpful in my experience. I think every horse develops his own "buttons."
Riding can be such an expensive sport! But maybe it's time you try to find a horse you can lease, or perhaps share, if you can't get one of your own.