If you haven't got any sockets with two live (hot) connections, a neutral connection and an earth connection, there is no way to safely run the appliance.
You should run the whole thing from a plug that can give you the two live (hot) wires and a neutral. It should be an L14-20 or something like that. All the NEMA plugs that have 120 V and 240 V have numbers that start "L14", I think.
Then you just wire the 240 V loads between the two live wires, and the 120 V loads between neutral and either of the live wires.