Can the DG6000 and the DG7000 run both 120v and the 240v at the same time?

You can use the dual voltage units on both voltages at the same time. It splits the available amps and watts to both voltages.

VOLTAGE SELECTOR SWITCH

This switch allows the generator to operate in either single voltage (120-Volt) or dual voltage
(120/240-Volt) mode. In single voltage mode only the 120-Volt duplexes and twist-locking
receptacles are powered. The full rated power of the generator is shared between the 120-Volt receptacles. In dual voltage mode both the 120-Volt and 120/240-Volt receptacles are powered; however, only half the rated power is available at the 120-Volt receptacles. Full power is
available at the 120/240-Volt twist-locking receptacle.

NOTE: Do not move the voltage selector switch while powering electrical devices. Disconnect all electrical loads before moving the switch. Failure to disconnect electrical loads can damage the switch.