When you are trying to move her, be VERY focussed on the part of her body you want her to move. If you want her to move her hindquarters away from you (always a good one to start with as ponies are usually better at moving the hindquarters than their forehand), look at the back leg nearest you, focus on it and kind of project your energy at it. Horses pick up on where you are focussed - so if you are, say, pushing her shoulder and looking at her ears, she probably won't be clear about what you are asking her to do.
You can lightly tap her leg or her hindquarters to reinforce - as soon as she tries to do the right thing, even if she only shifts her weight in preparation to lift the leg you want to move, immediately stop and look away - taking the pressure off is her reward. Then try again - try for one step, then quit to reward her. If she doesn't try, incrase the frequency of the tapping - NOT the force. But quit as soon as she tries. She'll soon get the hang of it.
But be clear what you want her to do - if you aren't, she can't be.
Nothing wrong with walking her with the dogs - you can build lots of learning into that. Just don't let her decide when she's going to graze. If you want to let her graze, give her a cue to say it's OK. I use "Easy" for my two, then relax my body, then when I want to go, I raise my body, focus on where I want to go and ask them to "walk on" - if they don't, the 12 ft rope is long enough to swing behind me and give them a tap on the bum, without me having to change my focus. It works much better that battling about grass.