Bicycle tire inflation

A question for my cycling friends about tire pressure. I have had two tires this summer where the bead of the tire tire has come unseated, and the tube has come out causing it to pinch and leak. The have both been on the rear tire. The were both Continental brand tires. The latest one, I bought a replacement tire that is a Continental Grand Sport Race tire. It says that its max PSI is 120. Before each ride, I air my tires up to their max tire pressure. Today, on this tire’s second ride, the tire came unseated, the tube pinched and leaked within my first mile. What I want to know is since I’m a big rider (250+ lbs) do I need to step down from the max PSI as I am assuming that my body weight on a the bike increases the pressure that is in the tire from the unweighted air pressure I measure during airing it up. Thanks