Double hauling is a way to get increased line momentum and by doing so. enables you to cast long distances and against the wind.
The secret of a good forward cast is a good back cast. When you cast backwards you pull on the line at the same time. This causes the line to travel at high speed and more imprtantly to put it into a tight loop.
The forward cast, again, is executed by a fast pull on the line just as you are about to release it for the forward shoot.
All this takes a great deal of explanation in print. Michael, I would advise you get a few lessons with a pro.
Double hauling can make fly casting on still waters a great deal easier, and less exhausting.
I have only the proper use of my right arm, and consequently have never used the double hauling technique. It hasn't prevented me from casting a line as far (and further) as the able bodied anglers I have fished with. It didn't stop me from getting into thr England Flyfishing Team and representing my Country at international level on three occassions, including the World Championships in 1986.
I suppose that what I am saying is that casting is really about timing as much as anything and unless you want to get into tournament stuff, using shooting heads and the like, going for extreme distance, who needs double hauling, which if done badly may impede your ability and would certainly give you tennis elbow.
Perfect your ordinary casting technique - you'll be able to cope without the rest.