The bench press, sadly, comes with a lot of anterior shoulder pain. You can tell someone to use dumbbells or a neutral-grip bar if they're suffering through shoulder problems, but some people would rather repeatedly get kicked in the crotch than give up the barbell.
If you're stubborn as a mule and don't want to swap it for a more joint-friendly alternative, start doing paused reps.
Most people don't use an optimal bar path. Ideally, the bar should go straight down and straight back up. If your bar path gets out of alignment, your shoulders can get out of position and round forward. When this happens, the head of the humerus moves forward which causes excessive stress across the tendons in your shoulder.
The paused bench press can help save your shoulder joint by improving the bar path and shoulder positioning. Pausing with the bar across your chest allows you to focus on guiding the bar straight down to your chest.
This prevents the excessive back and forth movement that generally occurs with touch-and-go reps. Eliminating excessive or unwanted movement will keep your shoulders in optimal alignment.
You'll also practice a better bar path because it'll allow you to focus on the eccentric, isometric, and concentric portions of the lift. You can better guide the bar straight down and straight back up if you're forced to slow the movement down.
Pausing with the bar across the chest teaches you how to stay tight and keep your shoulders retracted at the bottom portion of the lift. This is critical for shoulder health.
Another reason it works so well? It forces you to go lighter than you normally would. Like any lift that starts with an eccentric (lowering) phase, benchers can use stored elastic energy in the muscle to give them a spring back to the top. This leads to doing more weight than they can stabilize.
If pausing your reps doesn't help, try reducing your range of motion by lowering the bar to a board. You can vary the range of motion by adding more boards to accommodate your shoulder pain.
Generally speaking, I'm not a fan of stopping short of the chest without a target to aim for. Trying to pause your reps without a target to touch can become very imprecise and inconsistent. And people who stop short without a target to touch cut their range of motion considerably more on their heavier sets, giving them a false idea of what they can actually lift.
Using a board to eliminate unwanted range of motion will ensure each press is consistent in its depth.
For the paused bench press to work, you have to have a solid base of support. You need to pin your shoulder blades to the bench by pulling them down and back.
Even at the top of the press, your shoulder blades should remain glued to the bench. This allows your shoulder to be kept in a tight and packed position. Scapular mobility is a great thing for many lifts, but the bench press is not one of them.
Elbow tucking is another critical component. If your elbows flair out when you bench, using a paused variation will cause more harm than good.
When you lower the bar to your chest, think "break the bar." This cue allows your elbows to come into your sides instead of flaring out. As you press the bar upward, think "stretch the bar." This engages your lats which will lead to more stability through the shoulder.