Al, I live on the other side of the city from L.I., in northern NJ. The distances are much shorter than in Texas but the travel times are similar.
I have family in Syosset so I get to experience this from time to time, unfortunately. It's the 40 mile trip that turns into the 3 hour tour.
FWIW, I also have a slap tear - at least one that's been diagnosed, and that's my good shoulder- SLAP tear + arthritis. I have been experimenting with my press groove a lot lately in terms of whether it's more straight up or more of the open-the-door type.
The latter, in my experience, shifts a lot of the load towards the back rather quickly and, if you haven't tried it, may be something to look into. I didn't grow up bench pressing and my kettlebell press groove turns out to have very little carryover to the bench press because I've learned to rely on my middle and back shoulder muscles much more than those in the front and, indeed, the barbell move that I get carryover to is the behind-the-neck press.
I'm rambling but trying to say that you might find something that works for you and, if you do, my guess is that your two sides will look different as each finds its own best path.
Last but not least, I'd be remiss if I didn't point out that being able, even with bad shoulders, to press the 40 or 44 kg kettlebells is no small accomplishment.
-S-