What stands out to me is the way you are trying to get him to lower his head by putting your hands, with a rigid elbow, into a downward pointing orientation.
this, and the running martingale , are making him move with caution and defensiveness.
I bet he can really move out a a lot freer than that.
You need to get your hands to be almost "pushing" forward, rather than "dragging" downward. you want to encourage him to move his poll farther out and away from you, not higher and closer to you. So, really kind of visualize your hands pushing his poll out farher away. hold your reins more firmly , put your thumbs on top and position them so that your thumb points like a laser beam to the bit in his mouth, NOT toward the floor.
put a soft bend in your elbow and keep your upper arm close to your rib cage.
also, let your leg drape onto his side.
i would let him trot out, on a loose rein some, even if he giraffes. get him moving, in front of your leg, and happy to move, then slowly ask for some contact. if he puts his head up to avoid it, you follow him up, keeping your elbow, hand, to bit alignment, and keep asking him to accept contact. when he does, then think of pushing your hands toward his poll to encourage him to move his poll out more in front, and down, AND, put some leg on let him move out and into the bridle.
nix the martingale, if you can.