Read up on Eric Cressey. He goes in depth about all this stuff. I'm just a layman at the moment (come back to me in 4 years when I'm a physio haha) But I'm guessing you feel it in your shoulders cause poor thoracic mobility hampers good shoulder mechanics. So if you cannot get ample thoracic extension, your not going to be able to get your arms all the way over your head and the stress is going to move to your shoulder joints.
The test you did can also be a very gentle stretch. get in the laying down position and lay there with your hands above your head (trying to touch the floor - but don't force it) for like ten minutes whilst taking big deep breaths, eventulay you will loosen up.
I used to be the same as you and I just hammered the t-spine mobilitly.
I also do lots of forwards and backwards arm circles (like swimming) both arms at once, one arm at a time etc. great for shoulder mobility. You'll be suprised how knackering it is aswell.
Remember though, I'm only saying what worked for me, so don't force any range of motion and stuff