If you want an accurate diagnosis, go to a physiotherapist or chiropractor.
If you want exercises to improve your shoulder in general, Badminton insight on youtube posted a great one with 8 or so exercises. I can’t do the final one as it’s too painful lol. Most of the exercises require a resistance band thing. I got a nice set from Amazon with 5 different resistances and a door anchor and stuff. They work a treat when I can be bothered to use them.
My shoulder pain came on after I tried to do a hard smash when the shoulder was not in the correct position in the socket. Since childhood I’ve been able to slightly dislocate both shoulders, and this isn’t ideal for badminton because the angle where you prepare for smash/clear/drop is almost the same as where I place it to dislocate. I have to actually push it in a certain way for it to happen though.
3 years ago a Physiotherapist noted that my right shoulder was noticeably lower than my left one when I stand relaxed. No idea what caused it but it could have been from years of badminton. I also struggle to do lots of pushups. Used to manage 20 when younger, but now I can barely manage 8-15 before pain or exhaustion in the arm / shoulder.
There’s an exercise where you put your elbow as close to your hip as you can, and then move your hand from left to right (right handed), while holding a resistance band. I struggle with this one the most. Also struggle with wall angels.
With regards doing exercises, you could be doing them wrong. Or it could just take you longer to see a benefit from them. Don’t be like me. Do them as often as professionals tell you to do them.
This is the badminton insight video I was talking about. See if any of those help you out