Shoulder injuries are no fun, but it’s still possible to get a good workout if you modify.
For plank pose and pushups: Either place your knees on the ground or use the wall for lateral pushups/planks.
Pushups can be eliminated entirely by just holding plank pose with knees up or down. Both versions will strengthen the joint-hugging muscles around your shoulders over time.
When using free weights, drop to the lowest possible weights or even no weights and focus on form.
Modus shoulder exercises–plank, pushups, and free weights–have been carefully designed to be shoulder-friendly, but challenging to the surrounding muscles.