I’d have to either have a pretty good indication of where on the shoulder it is to say for sure (not that I can say for sure, being Not A Doctor), but you saying that it’s deep in the joint makes me suspicious.
With your shirt off, lift the affected arm over your head.
Grab that cord of muscle right in front of your armpit with a pincer-type grip.
Did you just jump out of your chair? If so, that’d be your pec minor, which shouldn’t really be involved with flat bench unless your form is shit, which you just said it was at first. Massage it, grip it tight with that pincer grip and hold for 60 seconds before releasing, heat it, ice it, and take 10 days off from benching while you do all this to pamper it and get it back to normal.
Dips REALLY inflame mine, I can’t do them without a pain deep in the shoulder “joint” but really the pec minor is just a muscle that feels deep.