Have you had anyone look at your pressing technique?
Do you understand the ideal start and finish position for the strict press?
Is your shoulder rom limited?
Do you have a stable shoulder position with armpit forward?
When you press is your back flat with ribcage down and neutral pelvis?

Are your shoulders internally rotated when standing?
Torque and Trunk Stability Part 1: How to Stand
Episode 274: The Standing Athlete

I would recommend not icing the shoulder.
Icing isn’t helping the issue either.
A resource on this is Gary Reinl’s book Iced