Here what I'm pretty sure you have going on.
Whoever wired your home originally assumed an electric dryer would be installed; maybe it was installed originally. They installed a standard 10-3 cable to the dryer location fed from a double pole 30 amp breaker. All that's fine. Then, someone decided to go with a gas dryer instead of an electric dryer and needed a 120v outlet. So, the 240v dryer outlet was removed and a regular 120v outlet attached to the #10 wires in the box. That is also fine but they should have changed the breaker to a 20 or 15 amp. It's not really that much of a hazard as the breaker is properly sized for the wire but code doesn't allow a 15 or 20 amp outlet on a 30 amp circuit.
Can you use this circuit for a 30A RV outlet? Sure. The existing 120v outlet is hooked between one of the hot wires and the white neutral leaving the other hot wire not connected. You can connect your RV outlet to this wire and the white neutral in the box providing a 30A 120v connection.
The only real problem is that technically your 120v outlet for your laundry equipment should be on a 20 A breaker. I would change out the double pole 30 A in your panel for 2 single pole breakers, a 20 A on the wire feeding the laundry equipment and a 30A on the wire feeding the RV outlet. Technically that doesn't meet code either because a multiwire branch circuit like you have is supposed to be on a single double pole breaker, not 2 single poles. However, that wasn't code until a few years ago, maybe about 2008 or so they introduced that. It's not really a hazard, just meant to make it easier for someone working on the circuit to get them both turned off.
