You have a NEMA 6-20 receptacle that is (presumably) wired with 12/2+ground cable. You absolutely cannot feed two 120V circuits with that. 120V circuits require a neutral wire, and it's not present in that cable.
You absolutely can replace the cable with 12/3+ground. This is a simple unit-swap to replace the cable that is already there, identical in all respects except you put the red wire where the white had been. The white wire becomes actual neutral. At the service panel end, it lands on the neutral bar. At the NEMA 6-20R end, it is capped off and unused.
Once this is done, the NEMA 6-20 can be easily replaced with a NEMA 5-20 dual receptacle with the "hot" tab broken off, or two 5-20 receptacles that both pigtail to the netural wire. This configuration is called a multi-wire branch circuit. Neutral must be pigtailed where 2+ devices are involved, and the breaker must be a 2-pole breaker where both "hot" legs trip and throw together. That is likely already the case. If you measure between the two hots, it must be 240V.
This will provide up to 1920W continuous to either pole.
Now, in this panel, neutral and ground may be tied together. Don't get overexcited about that. That means either this subpanel is miswired, or this main panel is the point of service. This is where the 3-wire (hot hot neutral) service from the pole has an equipotential bond to the house's safety grounding system, to return fault current and assure that transformer leakage or capacitive coupling doesn't make your 120V wires float up to thousands of volts. Anywhere but this service point, neutral and ground must be carried separately. They have separate jobs and one's job interferes with the other's. That's why you can't just bootleg ground for your neutral - because then you have no ground, and several types of ordinary wire problems could cause line voltage to show up on your equipment chassis.