STOP "trying" to keep your hands still. Instead, focus on relaxing and loosening up your hips, back, shoulders, and elbows. In order for your hands to remain still, the rest of your body must be relaxed and moving with the horse. Use your hips, waist, and elbows and "shock absorbers" for your body's movement. Keep your shoulders relaxed, allowing your arms to almost "dangle" from your body.
The only thing that should be a little tense are your abs. Tensing your tummy, JUST A LITTLE, (like you're going potty
), will help center your seat deep in the saddle. Tuck your bum under you a bit, open your hips and stretch your legs down and back (you may need to lower your stirrups a hole). You should feel a light stretch at the front of your hips/pelvis. Hold your stomach muscles just barely, while relaxing your back, shoulders, and elbows. It's a bit like skipping, chewing gum, and texting at the same time, but you WILL get it if you practice.
Once your seat is deep, your legs are back, and the rest of you are relaxed, you will find that you have solved the issue of your noisy hands! Everything comes from the seat. Once you fix it, many of your other issues will be gone.
I also agree that holding the head down shouldn't be your goal. If that's what your trainer is telling you, and he/she isn't helping you to quiet your seat and hands, then you should look for a new instructor.