Draw reins should be used rarely, and are more appropriate for a situation of correction rather than initial training. For instance, a horse that is extremely stiff and resistant in the poll as a result of years of bad riding. They should never be the only rein, a direct rein should also be connected to the bit.
They should not be used to bring a head down because this would defeat the purpose--the head should lower because the horse is stretching through his back and using his hind. Just forcing the head down does nothing for the back and hind. The headset is a product of proper carriage, not where it starts.
A loose training fork should be ok in the early stages, because if it is fitted correctly it will not be forcing anything, just preventing your horse from completely throwing his head away.
To get your horse to learn to use his back and stretch more--start on the ground. You can lunge in side reins. You should start low and loose and very gradually tighten them. Horses can panic and feel trapped when they first start this, so make sure to be ready to drive forward with your whip.
In the saddle, do large circles. Work your inside rein in big but gentle sweeps, with the same gentle tension you would use when opening and closing a door in time with your horse's stride, while supporting with the inside leg and bumping forward with the outside. At no point should jerking be involved and don't pull across your body, your movement should be slightly down and out. This will encourage him to stretch down. When he relaxes and drops his neck, reduce or stop the inside pull and just hold your hands level. Most often once a horse gets this message, they realize that stretching down is quite comfortable and begin to be consistent.
I would entirely cut out the jumping for now. He needs to be solid and calm in his basics before you do anything over fences. This will really counteract everything else you are trying to teach him.
If the high headedness continues and seems to be extreme, have him checked for mouth, back, or hip pain.