First of all, I am reeeeeeeaaalllly not a fan of the saddle. In general wintecs are crap and don't allow the rider to sit in a correct position and this one in paticular is too small for you and not allowing you to sit in it.
Fixing the saddle will help you to learn how to sit up straight with a vertical spine and equal pressure on both seatbones. I personally would have you on the lunge without stirrups or reins until you learn how to have a consistent, independent seat. Once you learn this, leaning will not be a problem. Also if you don't already, it helps to become fit off the horse as well. Pilates or yoga will both also help your core stength.
They say money doesn't buy happiness -- well happiness doesn't buy horses!