My wife uses her OLD cycling gloves in the gym. They did eventually tear the pads old of them.
Many guys in my gym use gloves. It is always for the same reason. Their wives and girlfriends don’t like it when their hands feel like sandpaper against their more sensitive parts. Keep wearing gloves. Your girlfriend will like it.
Old tire tubes cut up work good also. I also love chalk and batters rosin. However the batters rosin is so sticky, many gyms ban it.
BTW: If your rotators are weak (not Ruptured or torn) do rotator cuff exercises. Use very light weight. They can get stronger.