Thats almost a catch 22.
You can't be a better rider while you're a passenger, but you need to be a passenger to become a better rider.
At the beginning stages, you are working on your balance, rythm, relaxation, and following the motion. You are typically working on yourself as the horse carts you around. You're a passenger.
The next stage involves contact and a bit of influencing. What I'm hearing is that you want to be more assertive in the ride, but don't know how. Communicate with your instructor. She needs to coach you through how to do these things. They won't happen over night. A lesson horse who dives in with you is also diving in with everyone else. But they are safe to ride so the problem isn't fixed. You are asking advice on how to train, and you need to get this advice from your instructor.
The next time your instructor says "She's diving in", reply "How do I get her to stop?"
The horse is on the forehand because she is a lesson horse, and doesn't need to know much else other than to keep her rider's safe. So yes, you are a passenger to her. Ask your instructor if you could do exercises on picking her up off the forehand and making her use her butt more. Ask her how to halfhalt properly to bring her canter back. Ask her about leg yeilds. Aks her how to pick the horse up when she dives in. You want to start the "influencing" stage.
Forget about the headset. Ignore the head completely. You and the horse are not skilled enough yet to obtain a "headset" and when you are, the head will lower naturally.