OP--When you start to sit the trot with your stirrups...do you start to grip more with your thighs, making your feet come up out of the stirrups?
I find this is what happens to me if I pick up my stirrups. Before I start to sit the trot I make my horse walk/halt, and then bring my knees up to a 90 degree angle, swing them straight out away from my body, and then let them hang down. I found it helps me sit a bit deeper into the saddle after I have gotten the muscles to loosen up.
Keep relaxed through your hips and allow yourself to go with the motion. As soon as you tense up, you will start to bounce more.