No offense, but I believe you should get some professional help with your horse or maybe you could take some riding lessons? I understand that sometimes riding down-hill is hard, but you shouldn't fall off just because her head goes down. Lean slightly back when going down-hill and move with her centre of balance. You also said she has no withers but you really shouldn't be relying on the withers to keep you onboard.
Gripping with your legs is a big no-no. If she is green-broke, she might take that as an aid to go faster, or you could just plain confuse her. Heels down, move with her, and heels down.
Since you're new to riding in a saddle, I think it would be best for you and your horsey to get lessons. I'm sure you're not a beginner, but even experienced riders get lessons. And who knows, your instructor might be able to help you with some of your (you+horse) problems.
But I do think you should really work on ground-work with her.