Okay I am an advanced beginner rider. I know the very basics so this isn't coming from someone all that experienced. But this is what I know FROM experience.
You are probably also bracing against the stirrups since you mention "pushing your heels down"
The stirrups are only supposed to rest your toes, you stretch you LEG down, not your heels and you never push because then you create a block. Blocks prevent all the energy you get from the horse's momentum from moving down your leg and out your foot. Instead the energy stays on your upper half and that's when you get bouncy and slam around at the sitting trot.
So stretch your legs instead. Get your legs down and underneath you so your seat bones can fully function in keeping you on the horse.
Forget the headset. That is not dressage, and you're not baking cookies so no need for the cookie cutter ideas. You never pull a horse into anything. That's INCORRECT and creates bad habits. You push them forward, and capture that energy into a correctly timed half halt and allow the horse to find their own sweet spot. When the horse isn't bracing or resisting, they will be able to use their back properly, and their neck will come down and then all you need to do is just keep on riding the horse. Leave the headset to them.
In riding we work with the horse, encouraging them forward and around our leg or yielding to our leg. We don't force the horse by pulling them around or cranking the reins down.
Hope that helped :)