What do you mean re:

while I've noticed a huge difference adjusting my breathing, posture or pace while running

Like something you do in place of a warm up? or rather by comparison to a warm up?

I think, based on just layperson random reading, cool downs are not generally hugely important except that stopping abruptly can cause locking up (more uncomfortable than dangerous). Warmups however do seem to be implicated in injury prevention, I think it's a combination of gentle clearing out debris between the white slidey crap (fascia? no good with the terminology, just function) and the muscles + vasodilation/general circulation meaning they have a good supply of oxygen and don't do anaerobic work too early and tire out.

If you're not launching into flat out intensity it's probs not that serious though.