I can't be sure, but it sounds like you have a saddleseat saddle, which would make it even more difficult to post.
I do disagree with one thing, It doesn't matter if I'm in an english or a western saddle, I will ride the same way. If you ever look at really old western saddles (check out some of the vintage on ebay), you will notice that the fender comes directly under the seat, instead of the "chair pose" adopted by many of todays western riders with the stirrup in front. English saddles do feel different and do require the rider to have a stronger sense of balance. The 1" stirrup leather has a lot more give than the thick fender and strap attatched to the western saddle. If you have your feet too far out in front of you, and are trying to sit up straight, then yes, you will stand up, your feet will come forward and your butt will come back down with a thud. Many riders with feet to far forward or heels to far down (pressing feet forward) compensate by leaning forward with their upper body. Ideally, whether english or western, we want to line up shoulder, hip, and heel, have our seat properly balanced on the back, and legs under us, effectively on the barrel of the horse.
When you post, try practicing by giving yourself a hug at the standstill and stand up. With your upper body disabled with the hug, your legs and core will have to do the work. It feels a little harder when you can't use the momentum of your upper body to help you out. Make sure your pelvis stays level in the saddle and in the air when you stand. Once you get the feeling in your legs, try it again with proper posture. Notice that by pushing straight down into the stirrups you will gain stability.
If the horses center of balance is too far behind you, it may be a little harder to ride this trot. A good trainer and lots of practice will get you on the right track. Good luck.