Like you said... when you ride no stirrups your pelvis is more forward and your body is more back. If you do that with the stirrups it should help. It helps me to lean back and I almost never ride on my butt. Usually I stay on my inner thigh, as if I were standing over the horse, not sitting on it. Also, get your weight down. I find a good way to do this is to stand up in your stirrups, heels down.. then when you sit back down.. keep that same pressure that you had when you were standing on your feet.
I stop and stand every once in a while to get my weight down, but it helps.