The converse of using the back muscles is relaxing the arm muscles. My training method for this is to hold for long periods of time at full draw, like 20 seconds, which gives me enough time to become aware of which muscles are tensed and which muscles are relaxed. This is a practice exercise, not the way I usually shoot the bow, which is with about a 2 second hold. I started doing this with a 25# bow, and now am able to comfortably hold for 5 reps of 20 seconds with my regular 40# target bow, so it is also a strength building exercise, as well as an exercise in controlled breathing during the long hold time.
As far as your door closing with the elbow concept, I have heard the same thing recommended by several archery instructors. If it works for you, go for it.