The problem comes from too little pressure to expand the case mouth out against the chamber.
The solution is higher pressure loads, which possibly can be obtained at the low velocity by using a faster burning powder. Heavier bullets will help also, but bullet diameter with lead has little effect on chamber pressure. If you can purchase bullets large enough to fill the chamber to the closest fit that will feed smoothly you may solve the problem, and will certainly tighten accuracy dramatically. The accuracy improvement being because the bullets cannot lean as much while starting into the rifling.
Filling the loads with enough shot buffer to get slight compression will probably also solve the problem and allow you to keep velocities down where you want them. The compressed loads will shoot with higher pressure/velocity, which means you'll need to reduce the powder charge a little to maintain your desired 950 fps speed.
Shot buffer is available from any gun shop that sells shotgun reloading supplies and will be available in most larger cities, or try
www.ballisticproducts.com.