Pulling his head down will not soften his back. It works the other way. Get the hind legs, the back will soften and the head neck and poll will release.
A horse that does not have natural rhythm needs to learn this from it's rider. So YOU need to keep the rhythm with your body rather than just going with him. Sing jingle bells to yourself as you trot, keep you body at that tempo and rhythm, and he will learn to follow your lead.
Riding is like dancing with a partner. You need to move in harmony, but one of the partners takes the lead to help direct the other. You need to take the lead and guide your horse, without being forceful.
You don't need to use your reins, if you try to fiddle with your reins on this horse you're going to make the problem worse. Just ride forward, and keep your seat going in a set rhythm. If you let the horse dawdle, dragging its feet through the sand, you're not going to find his tempo and will therefore never find true rhythm.