Trying to jam your heels down causes stiffness right through th body and you feel you are bouncing because you are!
At the moment you have no seat and are relying on your hands for balance. Not good.
If I were teaching you I would have you ride without stirrups at the trot, reins in one hand and two fingers of the other under the pommel pf the saddle and pull it up off the horse. This will put you deep in the saddle so feel how you are sitting on your seat bones. Keep that feel when you let go and try to emulate it. Th moment you start bouncing pull yourself down again.
Whilst doing this let your legs hang down relaxed, you can, once sitting deep raise your toes but balance os the key to a good seat and the rest will follow.