Some additional thoughts on this: shoulder and neck tension imply levator scapulae/descending travezius use, muscles that lift your scapulas straight up and down. Basically, they make you shrug. You really don’t need to use them at all in the shooting sports, but they figure a lot in “tactical” stances.
Chances are, if you’re finding yourself in that super aggressive “tucked behind the gun stance,” you’re doing it right out of the gate with the draw. Provided that your holster is at the right height, you should be able to grab the gun and level it to your eyes leaving your shoulders relatively static, just using arm flexion/extension. Steve Anderson talks about positioning your upper body like a boxer on the draw — upright, shoulders back but relaxed — and I think that’s really helpful. This should make the draw faster, and allow for better control over establishing good grip.
That shoulder/neck tension can also screw with your index. Most people have a little bit of “drop” in one shoulder versus the other. If one shoulder comes up higher than the other under tension, it’ll throw the front site off in the same direction as the higher shoulder.