I do think the slow is part of the problem - we need to get to a "move forward and be quick-like about it" point. I need to get back in practice with carrying the whip - it's the double reins with something else in my hand that gets me all cattiewampus.
Another thing that might help is a pretty simple ground exercise to get the hindquarters to move.

Really all you do is apply pressure with one or two fingers to the hindquarters. Ideally he will move the hindquarters away from you, if his legs actually cross under and don't just side step you are really doing it right. Eventually the idea is move off the pressure smartly!
It is actually really good if you can get them to do the same thing on the front end and pivot on the rear. You might actually be amazed at how much it can help with just regular riding and manners. All I have to do with Lance is press with two fingers and he knows to scoot over. If he is really messing up (being a

) all I have to do is look at his hindquarters and walk towards them with a cranky look and he will move over. So much about horses is just simple pressure and release, not even physical pressure but body language speaks in volumes to them.