A couple of thing here:
First -- is this horse's conformation exacerbating the problem? Is he 'ewe necked'? Does he have a short, thick neck?
Second -- Is he worse when you 'take hold of him'?
Have you checked his teeth, saddle fit, etc?
A note on running martingales -- Don't condemn then without knowing how to properly adjust one.
If you put one at the length where it is in a straight line from the place where his mouth should be and where your proper hand placement should be, it is doing nothing at all. If in doubt, place it at the horse's throat when his head is in a 'normal' position. If a running martingale is lower than that straight line from the horse's mouth (when its head is in a good position) to the rider's elbows (not hands), then it has been adjusted too low. It should never pull down on the horse's mouth unless the horse's head is waaay too high. Then, they do not help so much with getting the horse's head down (this requires skill) as they help with control when a horse tries to bolt or spin around. The horse with his head in the clouds can be completely out of control and it can put a rider in a pretty helpless position. This is why race horses are galloped in running martingales. The jockeys could not hold them if they could get their heads too high and up into an out of a 'control' position. They never pull down on the race horse's mouth. A running martingale should not do anything when the horse's head is in a decent position.
As for bits, I would go to a three piece (French Link or Dog-bone) snaffle and I would probably put a nose-band (mouth closer) on him until you get him properly schooled. The schooling process will go much more quickly if you proactively remove one of the methods he will automatically use to show resistance to what you want him to do. Once the resistance is gone to your controlling his head position, the mouth closer can be removed with no problem. Gaping the mouth is just a symptom of the bigger problem, but it can make it a lot more difficult to fix the problem.
For technique -- You can get any horse's head down as low as you want it if you make him miserable enough when he carries it too high. Sounds harsh, but it is not. It is really quite simple and gets a 'happy' horse with good head carriage out of almost any resistant horse. It is the epitome of making the right thing easy and the wrong thing hard. [A skilled rider can get about any horse to travel with his head knee-high in about 10 or 15 minutes. If that is where he wants it. I have done this very thing many times in clinics and demonstrations.]
You simply take more hold of a horse's mouth as his head get higher and shows more resistance. You take more hold while you 'push' a horse forward into the bit. If you take enough hold, he will get desperate and will try to 'hide' his head somewhere. Every horse will put their head down at some point to get away from your firm hold. The instant his head drops, you give him a release of all pressure on his mouth and sides. For a while, his head will go up and down and everywhere trying to figure out where you let him be 'comfortable'. If a rider is 100% consistent -- making him pay a high price for elevating his head and offering release and comfort for bringing his head to a acceptable place -- he will chose to carry his head where you want it. This is much preferred to forcing him to carry his head where you want it. He will chose that place because that is where you let him be comfortable. That is where you take the uncomfortable pressure off of the reins and 'reward' him with that relief.
You DO NOT need to 'praise' a horse. That is totally for your pleasure and not his. The only praise a horse needs is for you to take the pressure off. Other forms of praise come long after the deed causing it is forgotten -- long after a horse's immediate memory. A horse will literally do anything he can to get release of pressure -- defined as a firm contact with his mouth and/or his ribs. That is HIS preferred reward.