Riding without stirrups is definitely a good idea.
Riding small grids and gymnastics without stirrups and perhaps reins is even better. As you do the exercise, really concentrate on 1.) waiting for the horse's jump to push you out of the tack, 2.) staying over the middle of the saddle and 3.) feeling your hip move.
Often the release is part of the issue with jumping ahead. If you're tense and your elbow or shoulder is locked, releasing pulls your upper body forward. If your horse is managable, knot your reins and once you have a good approach to the gymnastic, drop the reins and let your arms dangle, or experiment with different hand positions - airplane arms, hands on hips etc. Then do the exercise again with reins and your usual release and see what changes. If the jumping ahead dramatically reappears when you pick your reins back up, do some work on relearning your release - jump through the grid w/o reins again and practice mock releases, learning to move your hand and arm without it pulling your body forward.
If the problem is more a stiff or locked hip, try the rider's push up.
With your instructor or a friend standing at the horse's head, find a good, balanced two point where you can maintain your balance without support from the neck. Make sure you're over the middle of the saddle, with buttocks almost but not quite touching. Then slowly fold your hip until your chest touchs the horse's neck - make sure you keep looking up between the horse's ears so you don't fall forward!
Then, without pushing or balancing on the neck, slowly come back into two point. This is harder that it sounds! Repeat 10 times, focusing on balance and correct position - make sure you're folding the hip and keeping your back flat, not bending at the waist. When you've mastered the exercise at the halt, do it at the walk and trot if your horse is steady enough.
Hope that helps!