Might be a SLAP lesion; might not. You might have one but the pain could still be related to bursa or the rotator cuff and proper PT would be the solution.
The diesel crew shoulder prehab series is as good as any approach - sight unseen. Some folks swear by KBs for rehab, some by low-weight OHS and controlled kipping pullups (of all things). A few thoughts:
Without or after surgery, you first need to get the inflammation down to effectively rehab it. This means avoid the movements that consistently cause pain, including rehab exercises or favorite lifts that cause pain. Rehab can hurt sometimes, but if it consistently hurts an injured area, you're probably not doing yourself any favors.
I'd stay far away from bench press and dips and the surrogates you've been doing excentrically, and I'd avoid them until well after you can overhead lift comfortably. Also, stay clear of flies and, though you didn't mention them, upright rows. If you have an impingement, these are likely to aggravate it and don't have much benefit anyways.
You may simply have to accept that you'll lose some chest and shoulder strength as you work through this. The fact of the matter is that your raw pushing strength is currently greater than your ability to comfortably stablize it. Get the inflammation down, work your mobility (stetching/, improve scapular stability and rotator cuff strength, and then rebuild pressing strength. In the meantime, you can do unilateral work on the uninjured side. It can have more carryover than you might expect.
The general rule of thumb when rehabbing a shoulder injury is to work 3 pulling exercises for each pressing exercise. Now this rule of thumb is based on a population that spends its days hunched over a computer or steering wheel. If you dig ditches for a living, that ratio may change depending on how much pushing and pulling goes on between 9 and 5. Scapular mobility and stability/strength are key to effective shoulder function. There are tons of exercises to choose from. Hafe at it.
In terms of pressing exercises, you'll want to concentrate less on maintaining bulk strength at first and instead work the stablizers needed for the pressing movements. You can gradally add more demanding pressing work as you discover variants that can be done pain-free, a list of things that will grow larger as inflammation decreases.
In terms of reintroducing exercises, start working limited ROM before full, isometrics and slow speeds before full speed, unstable (stablizer biased) before stable (skeletal biased). For example, you might start with pushup variants, incorporating isometric holds and fewer reps. You might work slow pushups with your feet elevated on a stability ball before attempting regular pushups. The next step might be neutral-grip DB floor presses. You might find that barbell board press work before full ROM DB bench presses (or not). Either would come before a close-grip bench press, which would come before a standard bench press.
You'll need to find your own way through this. If it hurts, try something else. I was able to go to overhead pressing well before I could bench and incline bench has been a no-no for most of my adult life. Other folks can bench before they can lift overhead. The shoulder is a complicated joint with many degrees of freedom, so there's a long list of things that can go wrong, and diagnoses like GIRD or even SLAP tear might not adequately describe the manifestation of your particular issue. One thing for sure, if your overhead form is messed-up because of the shoulder problem, you aren't doing yourself any favors pushing on with the training program. Get the problem fixed, maintain