I kinda have to disagree with your first sentence there a bit. Many times when I was under extreme pressure in a tournament match --- what got me through was taking it one shot at a time and focusing on my technique.Thinking about technique instead of focusing on achieving a result is only for practice sessions.
If getting draw is the objective, turn that outcome over to the subconscious and avoid conscious intervention.
If the mind feels the elbow drop is right it should understand driving down and through to maintain a draw plane.
As Brian noted everyone has a preparatory phase in which the whole of your being is readying to produce a result.
Where every body part from eyes to feet to hands and the extended self of the cue, are located is determined by what the body decides based on what it needs.
It’s a marshaling of resources, not a preprogrammed list from a control obsessed conscious mind.
Your comment “ When I try a draw shot without dropping my elbow, I notice that even though the contact point is not necessarily really low, I tend to miscue.” is a description of what happens when the conscious and not the subconscious is driving the bus.