Shaking or feeling unstable during a heavy bench press has two main causes: 1) you’re not using your legs to provide stability and 2) your core might be too weak to help stabilize the body during strain. Think about pushing your feet into the floor and driving your knees out throughout the entire press to get your legs involved. For your core, add things like planks and heavy carries to improve the stabilization strength of your core.
Source: Shinkle, J., Nesser, T. W., Demchak, T. J., & McMannus, D. M. (2012). Effect of core strength on the measure of power in the extremities. The Journal of Strength & Conditioning Research, 26(2), 373-380.