I don't like cavaletti with gaited horses, their gait is supposed to be fluid and smooth. Cavaletti to me is not the way to teach horse to watch feet or gain height.
It could be more than anything your riding mechanics as you may be letting horse get too strung out. And you could have your saddle too far forwards and are impeding shoulders too. Big problem with gaited stock, as well as other breeds too.
And would not be a bad idea to have him looked at by lameness vet either. This dragging could have root in physical problems.
And the running walk is not a high knee action gait either.
Working horse is plowed fields with soft dirt would build up strength better than cavaletti will.
And if you want more lift, you could try roller beads, which are wooden and lightweight, or even go buy lengths of the light plastic chain like you hang flowerpot from as even those will work. Secure with the same thing used on small bird toys, that twists shut? Thing is the movement.
Horses make me a better person.