You probably won’t like what I am going to say, but the first thing you need to do is get rid of the draw reins. They actually make the wrong set of muscles stronger – – the horse is not correctly stretching to the bit. He is actually using his neck to get behind the bit. You don’t say whether you own, lease, or are allowed to ride this horse. This can be fixed, but it will take a rider who knows how to retrain the horse. The “super heavy on the forehand” and the lack of engagement from the back end are typical of a horse who is badly behind the bit.
Think of it this way: the forward impulsion comes from the horse pushing himself froward from the rear end, with that energy flowing through the horses body and being caught, controlled & directed by correct use of the contact the rider has with the horse mouth through the reins. In order to do lateral movement, the rider must be able to direct the horse’s energy sideways, which is pretty much impossible if the horse is always behind the bit.
You either need a new instructor who can teach you how to correctly put a horse on the bit, or some one who can retrain the horse, or a different horse who has already been taught to correctly work on the bit.