You will find that many a Morgan, even those that were not trained in saddleseat style riding, are very good at “faking” correct carrying posture. Morgans already tend to be fairly upright in their neck set and so carrying themselves that way comes naturally. Saddleseat does not tend to encourage the horse to round through the back and lift the base of the neck, but because of the natural posture Morgans are built to take, they can “cheat” easily.
I’ve restarted a few Morgans: I really don’t touch the reins much for a while. I don’t even want the horse thinking about putting his head in a particular place. When I pick up the right rein, the horse should soften his jaw to the right and release his neck down and slightly to the right. When I pick up the left rein, the horse should soften his jaw to the left and release his neck down and slightly to the left. Until the horse can gets this really clear, I don’t pick up two reins. If the horse braces when I pick up one rein (which they often will do) I wait at that pressure until the horse releases and relaxes his topline.
You’ve also already picked up on my next suggestion, which was lots of circles and very few straightaways for a while. Circles are excellent because they allow you to use the one rein to give the horse a very simple feel to follow - out and down, while allowing him to find the inside hind and rebalance himself. Straightaways offer much more ability to rush and get heavy. If your horse has plenty of forward, it’ll still be there when you need it, but right now his desire to go forward largely anxiety-based as he doesn’t really know how to balance himself.
I also try not to let any horse at this stage go too long without some kind of transition. You’re in the phase of breaking old habits and ways of going, which means you need to counterbalance those habits with requests and small successes at new ones. I don’t criticize the horse, but I am particular: he doesn’t get to make up transitions by pulling off his front end and bringing his head up anymore. He doesn’t get to halt by leaning on the bit. He doesn’t get to take a few nice steps of balance, relaxed trot and then start rushing like a freight train. Interrupt every attempt at an old habit and make the request for the new one, reward every try no matter how small and you’ll make very fast progress.