Yeah, I would get yourself to an ortho if possible to get checked.
Overhead pressing movements, in particular, if shoulder gets tight can lead to some impingement of hte posterior labrum and possibly tearing.
Although it could also be SLAP lesion.
Is the pain more anterior or posterior inside the shoulder?
Does an internal rotation stretch such as sleeper stretch hurt?
Either way, pain inside the shoulder is usually indicative of labral issues so get yourself to an ortho.