I hold the ball about chest high, speed my feet up a bit, and drop the ball to the bottom of my swing on the third step rather than focusing a whole lot on a pushaway. This gives the ball a natural backswing height rather than me having to help it a little when I was holding it waist high.
It also helps to think of it as starting over a bar about 4" higher than where the ball is in your stance (it doesn't matter what height you have the ball as you will lower it later for slowing the speed down in future lessons) and push the ball over the bar and let it drop into the swing. If you looked at this push away from the side it goes slightly up then drops into the swing it is more of a rounded shape, it feels pretty vertical though so if that is what it feels like then you are doing it properly. The slight push up helps with timing since there is a natural pause even for a slight split second when the swing starts, that the slight upward motion will help set the swing in better time with your feet.
You can practice this at home by getting the ball in your stance and step with your right foot and push the ball slightly up, then back into your stance then step with the right foot and push slightly up then let the ball drop into the swing then repeat it again.
The balance arm will take a while to get used to, mostly because as soon as you stop paying attention to it it will revert back to where it was, so stay on this as long as it takes, the same with your trail leg.
It sounds like you found a good coach, work on these things, take them one at a time and make sure you do them all the time without reverting back. Let me know if I can be of further help, good luck.