Search for internal shoulder rotation stretches on here, like the banded bully. Odds are your shoulders are translating anteriorly causing the issues.  Look at your shoulder position to see if they are rounding forward. Sounds like missing internal rotation and extension to me.

You don’t have pain overhead because these positions are 2 different shoulder archetypes.  OH work falls into flexion & external rotation while dips & push-ups fall into internal rotation & extension.  

Make sure your position and load ordering is perfect like Kaitlin mentioned too.

Hope this gives you a place to start.

Cody