I messed up my shoulder and had to have surgery. Now I'm smarter about it. Same with bench press. He told me to even just put a little 5-lb plate on my chest, or 5-lb plate on the ground for pushups, and just go to that. Going all the way down so the bar hits my chest will cause more shoulder problems as I age (I'm 53).
Also, I spread my hands out too wide before. Not any more.