Had to stop doing push ups in my mid 50s because they caused my right shoulder to feel like it was becoming unhooked at the collar bone. Torture. Had been doing them since I was 15 along with weightlifting. Weightlifting still doesn't bother me at 72. Don't know what it is about the specific motion needed for push ups.
What is your arm/elbow position (elbows splayed out vs flush with torso)? How far in/out are your hands?