One thing to watch with overhead pressing is form. People don't think about it too much because they think overhead pressing is rather straightforward. But... the truth is a little different. Form is critical. I actually hate seated overhead pressing with a barbell, you really restrict yourself from doing the movements you need to do when pressing a large weight overhead. Standing military pressing is better, in my experience. When you do seated presses, people tend to push their body back into the seat and try to remain fixed with just their arms moving the weight. This isn't how it should happen. For example, your head should move forward through the "window" your arms create as the weight goes up. Make sure you do this after the bar has passed your chin, slamming that mother into the bottom of your jaw as it goes up does indeed hurt. Lyle Mcdonald wrote up a good bit about military press but I can't find it at the moment. I'll try to dig it up and post it later.
Push presses are great too. I also like doing face pulls at the end of my workout.
But why does it seem like so many older guys have shoulder issues? Obviously some of that is just age. Our joints aren't as fluid and everything isn't as flexible as when we are younger. That's the same reason why young children can sleep on a floor rolled into a position that would leave any of us hunched over and crippled the next day.
However... I think some of this is too much emphasis on pushing movements. You've heard this a million times, I'm sure, but your back, chest and shoulders need to be balanced. Do pulling exercises in combination with the pushing exercises. Another thing is that a lot of people try harder on benching then take it easy on rows because rows are hard. You need to pour equal effort into both sides of the body. If you have more chest and shoulder exercises than you do back exercises, you need to evaluate your routine.