Ah, yes, the “boulder down a hill” running away with you slowly routine. Its a draft cross classic. (look up the blog Riding With Scissors, it will make you laugh)
Bitting these guys can be a challenge, I have the same cross. Not just to remind them you are there but because their mouth shape is often a combination of thick tongue and low palate. You might want to try a straight bar snaffle (mullen) or a mullen tongue relief snaffle like Myler’s or the misnamed Sprenger Correction Snaffle. I use the Sprenger for training and it is fantastic. I had similar results with single joint and french link.
An alternative idea is to ride in an old fashioned short shank steel mullen pelham with double reins. Knot the curb reins so you have an emergency brake, school via the snaffle. I would hesitate to go directly to the kimberwick on green horse, someone did that with mine before I got her and its been tough to teach her forward and not hiding behind contact.
When you get the running away with slowly routine, halt and quietly back up a few steps. Then, halt and ask for trot. Reward the moment of forward and light, reward any lightening in downward transitions, backup and and push into trot on the boulder-like transitions.
Lots of patience, its SUPER hard for them to build the strength for this. There’s just so much horse to balance