I GOT YOU BRAHHHH. Look up coracobrachialis stretches! Its a muscle that runs from the coracoid process (on your chest/shoulder right below the ac joint) to the middle/outside of your humerus. It is used to pull your arm towards your torso and to stabilize your shoulder joint and it is especially active when you do incline shoulder/chest workouts (it adds to the pushing strength). It is also at its max strain when you are coming down from a chest press (probably starts at around 1/4th of the way down). One of the most stressful things for this muscle is when it contracts as it lengthens (as in exactly what u were describing with the chest press).
One of the simplest stretches for it is to stand up straight, good posture and raise your arm to the back and side diagonally about 45 degrees. You should feel a stretch on your chest/inner arm and possible a stretch or release on the tricep (by the way, its pain referral pattern is exatly what u described, rear delt/upper tricep).
You can look up all of what I just said online to confirm. Do the stretch I mentioned and hold it for about 5 secs as you feel the stretch. Repeat the stretch about 15 reps for each arm (YES BOTH). And check how you feel. I bet its a ton better.
Depending on how strained it is, you might need to rest/stretch it for a little (days to a couple of weeks) while and DEFINITELY start with really low weight exercises that use that muscle.
ONLY AND ONLY when you can do chest presses with minimal weight, full range of motion without any pain, you may raise the weight by a little at a time. If you dont have patience and let it work itself out, you will never leave this state. Check your ego and youll be gtg in no time.
For the mean time, focus on other body parts (heavy leg workouts and such help your entire body to heal and grow through hormones) and especially the antagonists of this muscle (think dumbbell tricep kickback) with low weight as well to help it release and balance you out.
Approach this with care, thoughtfulness and dedication and youll be pumping iron in no time.