Lifters suffer from a lot of anteriorly directed shoulder pain. It happens when the head of the humerus is pushed too far forward in the glenoid fossa. It feels awful.
This position is exacerbated at terminal horizontal abduction, elevation, and external rotation, which are all components of traditional bench press variations. But this can be avoided by using a squeeze press, even when you're managing some nagging shoulder pain.
Use dumbbells and squeeze them together as hard as you can throughout the concentric (lifting) and eccentric (lowering) portions of the lift. You'll get a legit training effect without flaring up your shoulders.
By forcing your shoulders and deep humeral and scapular stabilizers to initiate tension into internal rotation and depression, the head of the humerus will translate more posteriorly and will be more centrated in the shoulder socket. This positioning allows you to use internal tension and torque production through the shoulders and still train hard.