When wiring this with a dual ground, do I connect both grounds to the single ground post?

This outlet has no ground terminal, it is strictly for existing range circuits pre-1996 NEC. This receptacle accepts the red and black hot conductors and the white neutral conductor. In pre-1996 NEC installations, the ground strap inside the wiring compartment of the range must be jumpered to the center neutral connection post. Grounding of the range is through the neutral conductor. This receptacle is considered illegal today unless grandfathered in an older home. 1996 and newer range circuits/receptacles MUST be 4-wire.

White is COMMON, not ground. This outlet does not have a ground connection (you can connect the ground wire to the metal housing if you want). But the best thing would be to get a four prong outlet and a matching pig tail for your stove.

Yes that's the way to connect the wires.