I pressed an 82nd last summer a few times but didn't notice this happening. Did you have the limbs backed out on your bow? The one I worked on was maxed out. You might be able to make a short piece of parachute cord for each end that would go from one side of the press brackets to the other side of the brackets. This would go around the limb pockets and prevent the press sliding up the limb.
Maybe a better fix would be some heavy velcro, I doubt it would take much to hold it.