I have been jumping for a while now but only at a posting trot. I really want to move on to a sitting trot and then a canter.
It is extremely reckless and borderline negligent to allow someone to jump before they have all of the gaits down! I can't tell you how many horses I have ridden that will trot a jump and then immediately break into the canter upon landing. You need to have full control of the horse before you take them over jumps and if you cannot properly sit the trot and/or the canter... you have a lot of work to do I am afraid before you're ready to be jumping.
Where I ride, no one is allowed to jump until they have mastered all of the gaits with and without stirrups. Then they have to demonstrate that they can two point around the ring a few times including once without no stirrups - without losing your balance once which is a lot harder than it looks! Only then are you then started over the tiniest cross rails and it is a slow progression going from there. It's not just for your safety but also the horses safety.
... and the fact that your instructor put you on a green / lesser experienced horse with your limited experience is insanity. Like I said, this is nothing against you but your instructor clearly can't grasp where you are in your riding ability and is probably going to get you seriously hurt again. If there is another barn near you, I would strongly consider taking a look with them and talk about some of these concerns so they know not to repeat them.