The reason your robot is not functioning correctly is not weight distribution. The problem is that your gear ratio is too fast for the weight of your robot, and the torque of your motor configuration. Adding more motors to your drive could help, but as a rough estimate, I would say it would take all 10 of your motors on your drive with a very light robot (drivetrain only and made out of aluminum) to be able to get your 1:5 gear ratio running on the mechanum wheels.
In short, a 1:5 ratio on your drive with mechanum wheels is just not going to work on a typical VEX competition robot, especially one that is made out of steel. I suggest changing this ratio to a 3:5, changing your 12-tooth metal gears on the wheels to 36-tooth plastic gears.
~Jordan