I used to do the exact same thing that you do. And I have a crazy bouncy horse, as well.
What I was told was to think of posting as the horse bounces you up, and you use your leg muscles to slowly put yourself back into the saddle so you don't hit hard. So try focusing on going down gently rather than pushing yourself up. It's also easier if your leg is just slightly farther underneath you. Try standing up in your stirrups, not holding onto anything, until you feel balanced, and let yourself sit down without moving your legs at all. This is where your legs should be because it keeps you in balance atop the horse.
I think everyone has really good tips about riding without stirrups and working on the sitting trot more. Everything having to do with the trot ties in together somewhere. It's just lots of muscle and hard work. Also, the lunge line lessons are a fantastic way to learn things if your horse knows how to lunge. My new trainer doesn't do them, but my old trainer did and I used to be terrified of them but now I miss them and wish I could do them because it helps SO much with your position and everything because you don't have to worry about the horse.
Also, if you can, It might be a good idea for you to ride a different horse to just get the feel of posting on a horse with a not-so-bouncy trot so that when you go back to your horse, you can feel confident that you know how to do it correctly, you just have to figure it out with that specific horse.