I agree with first response - video of benching to confirm form is proper.
Once form can be ruled out as a potential issue...
I used to have a similar pain/issue with my shoulders from benching.
1. I changed benches to get a wider, firmer bench at 12" from my original "softer" bench. I still use it now.
2. Banded face pulls, rear delt exercises, also incorporate some reverse bench work with bands if you can. Anything to work the opposite muscle group (think bicep and tricep in your arm).
Good luck - it took me months to figure out my bench was the biggest issue.
My 1st post
I'll just chime in with the others that delt work has saved my 41yo shoulders. (and good form)
I start every training session with 20 lite lat pull downs, 20 face pulls along with 10 external and 10 internal rotations, and 20 lite reps on with DBs and the shoulder horn. It goes pretty quick (under 5mins) and keeps the pain away so I can bench and OHP pain free.
I've been doing this at the start of every training session for 5ish years. Every once in a while I get the idea to try and drop one or more of these and give it a few weeks my shoulders will start bugging me.