Yes.. put draw reins on and see-saw! It will work really well until she finds another way to evade contact, like say tucking behind the bit and backing entirely off.
I would continue what you are doing, lunging with side reins. When you are riding, work on shortening your reins, keeping your elbows on your hips, your hands down, soft and quiet and getting her to connect more "over the back" into your contact. Remember that the leg must always support anything done with the hand, and for the love of Pete be grateful that you have a horse that seeks the contact!! So so so so many horses are so backed off of the contact from people see sawing, putting horses in draw reins and ramming them in the mouth whenever they stretch towards the contact that it is no wonder we are starting to see this as the norm. In dressage, we want the head on the vertical, or slightly in front!!! This shows that the horse accepts the bit and stretches towards it.
Lessons are also a good investment. A pair of experienced eyes on the ground are so valuable. Good luck!