My mare is half draft, talk about an eating machine! I took a natural horsemanship approach with her, doing join up, etc. and it took a while, I was starting to give up. she would try to walk up to me on the lunge line, she wouldnt want to stay on a circle, and she wouldnt want to go faster than a trot. I was also a person hesitant to use the whip, but you have the whip there for a reason. I know its hard to get used to, but they have to learn to respect the whip, and if that means you have to hit them a couple of times with it, so be it. That doesnt mean whale on them with it, but give them enough of a smack that they respond and understand that the whip is something they must respect, not ignore.
In the round pen she would do like you say, stop and eat grass as soon as I stopped and tried to get her to come in to me. If she started eating grass I would push her out and work her again. I thought it was never going to work, I was starting to give up, but one day she finally came in and stood in front of me, not even putting her head down once to try and eat grass. We had finally "joined up" and I will tell you, it was such an amazing feeling. Now, I can simply stand in the middle of the round pen and hold the whip and she goes by voice command mostly, with a little whip encouragement at the canter (by encouragement I mean holding the whip up, maybe cracking it, but never having to touch her with it)
She only eats grass when I tell her she can put her head down, and she follows me everywhere without a lead line because I am her "leader"
Hope this helps!