hmm, i've little western training experience, but you might see if your horse was trained english? Shaking head probably mean something is wrong with the bit or teeth, maybe have someone come out to look at them. A kids-friendly horse should surely be able to stop, so i'm not convinced that this is a training issue. I'm guessing it's a health issue, but it's so hard to tell over the internet =P
As for leading, just practice LOTS. It's harder to training horses that lag behind than those that rush forward (for me). I would just use very gentle hands and offer release from pressure as soon as the horse moves. Maybe the horse will eventually stop lagging, but it'll take some time =) My best training tip is patience!
If I were in your shoes i'd get a vet check or someone to see if there is a health problem. It sucks that you can't find out more about the horse, but that's life. I know next to nothing about my horse, except that she's been reg. Oh well. I wish you the best of luck finding a solution =)