I experienced exactly the same injury as you three months ago. Whenever I did any sort of chest exercise, I would get a sharp pain on my right shoulder but any other excerise I did, my shoulder would be fine.
I went to a doctor to get it checked. He told me I had a small tear on one of my rotator culf muscle. And what caused the problem was that when I bench pressed (flat, incline or decline), when you are coming down with the weight you want to keep your elbows at 90 degrees and make sure you dont go past that point. If you do, you put alot of stress on your shoulders. The trainer at my gym also agreed with this. I guess I was putting too much stress on my shoulders.
So to get rid of the pain the doc told me that I needed to build muscle around that teared area to support it. Man I dont know the name of the exercises he told me to do but I will try my best to explain them. I used adjustable cables to do these exercises. First is to lay your arms to your side and grab the cable weight one arm at a time and bring your arm up 90 degrees in front of you. It’s a pretty standard shoulder one everyone does. Then for the second exercise, turn yourself around and do it in reverse. Grab the cable and pull it behind you until you cant pull it back. Usually I could go around 30-40 degrees before my tricep kicks in to stop it. The third one is have your arms rest stright against your body again and bend your elbow 90 degress front of you and grab the cable and pull it from left to right rotating your shoulders. Make sure your elbow is against your body the whole time. This is also a very standard one. And the last exercise he told me to do was the same as the 3rd one but instead of going left to right, you go right to left. I did 3 sets/10reps of all four exercised i mentioned above.
I’m sure you can do other shoulder excerises but make sure you dont do ones where you put weight on your shoulders. For exmaple lifting weight above your head back and forth putting tons of weight on your shoulders.
Also do not do ANY exercise that hurts your shoulders for awhile until you build up strength around the area that has the tear. I didn’t do any chest lifts for one month. And everytime I hit the gym I would work on my shoulders. Now I am back to my routine chest lifts and I feel no pain whatsoever.
Hope this helps.