Similar issue for 6+ months. If I over do most upper body work (pressing/flyes) especially, shoulders quickly become inflamed. What has worked for me are prehab/rehab - google 7-minute rotator cuff solution. I do suggest work out both as suggested on upper body days and very light with 30-50 reps on lower body days (gets circulation into shoulders to aid healing).
The 7m solution also suggest changes to some exercises, which I’ve done and helped greatly.
Supplements are fish oil/Flameout, curcumin, and antioxidants (ALA, NAC).