You could try a neck stretch they are like draw reins except there stretchy so if your horse needed to pick up its head it could and also there more relief for the horse.
Maybe try a loose ring or 3 piece to allow more freedom. A martingale could also help her head stay down but start off with it loose if she hasn't had one on before.
D-rings can be very stiff and not allow for much freedom in the mouth which might be why she feels unresponsive. I would try a single or double joint loose ring snaffle. I might just be partial because I swear by loose rings and naithe bits. The joints and the loose ring enable more finesse of your hand and promote suppleness of your horses mouth.
As for her giraffe-like tendencies, training aids help teach the posture, back strength is necessary for her to be able to carry you while on the bridle. Keeping her on the bridle requires connection of hand and leg.
You could try a rubber snaffle and do lots of flat work. You can also ride with draw rains, a bungie or chambom. Try doing a lot of lateral movements and work at the sitting trot without your stirrups some might be helpful. Good luck! Have fun!