Pain tells you something isn’t right.
It is preventing you from doing certain skills/elements. It needs attention.
Has anyone watched your technique to determine if there is a technique error?
What is the rest of your body doing when you perform chest pressing, DB presses etc.?
Shoulder or up/down stream deviations in technique or positioning can be where there issue originates.
For the time it is not preventing you doing the movement, however, that can change anytime.
Reaching your body’s tolerance for the deviations in technique will shut things down.