I think you need an entirely new circuit for 30 amp 120 volt use.
I am not experienced enough to recognize the box you showed or for what purpose it was installed. Does it currently power something other than your RV? If so, it may not be available for RV use.
The wire gauge looks small. You said each leg has a 20 amp breaker. That means the wire gauge should be 12 gauge for no more than 50 feet. It looks like 14 gauge which is only good for 15 amps. If the distance to the service entrance is longer than 50 feet, the wire needs to be larger (maybe 10 gauge).
Since there are two separate breakers, it is not wired properly for 240 volt use. 240 volt use would require a two pole breaker where the two breakers are mechanically linked so if one blows, the other will trip also. However, there may be 240 volts present.
Measure the voltage from black to red and to white. White to red and white to black should each read 120 volts. Black to red may read 240 volts or zero volts. Either way, you can correct the ground to neutral wiring and connect black and white to one 20 amp outlet and red and white to another 20 amp outlet. You cannot use the red and black together.
Ground must be separate from the white neutral wire. I don’t know if the ground requires an additional ground stake at the box.
You need a whole new cable back to the service entrance. The existing wire is too small. You do need a new box and socket for a 30 amp 120 volt outlet.
None of the existing wire or box is usable for a 30 amp 120 volt outlet. You need 10 gage wire back to the service entrance. If the distance is greater than 50 feet you need even larger wire.
You need a new 30 amp 120 volt breaker in the service entrance to connect the 10 gauge wire to. You need a box to house the new 30 amp 120 volt outlet. It would be good to have a 30 amp breaker in the new box to disconnect or reset when in use. Again, I don’t know if you need a new ground stake at the new box.
Note: If you hire an electrician to do the work, don’t assume he knows what you need for your RV. Electricians who are familiar with other home wiring have sometimes assumed the owner needs 240 volts. You do not. You need 120 volts. Several people on iRV2 have reported frying all their electrical when they plugged into a new 30 amp outlet that turned out to be 240 volts.
I wish you good luck and happy trails ahead.