With permanent magnet DC motors at a high duty cycle (or constant on), if the motor is turning slowly the current is much higher than when they are turning fast. So, if you load up the motor and force them to turn slowly many will overheat and potentially burn out due to the excess current.
It sounds like this is what is happening with your system - the motors do not have enough torque to get your device moving fast enough to allow the motors to spin at their rated speed. You can change your gear ratio to allow the motors to spin faster or you can get more powerful motors.