First, stop doing pushups for a week. I know, blasphemy. But your shoulder needs a timeout. Instead, try wall angels—stand against a wall and slide your arms up and down like a dorky angel. It loosens the shoulder blades.
Second, grip the floor. Yes, grip it like you're trying to twist it off. This activates your forearms and pulls your shoulder blades together. Suddenly, your shoulder feels like it has a support team instead of a drunk uncle.
Third, do pushups on your knees or an incline. It's not cheating—it's intelligent geometry. Your shoulder gets a lighter load, and you build strength without the crying.