Without knowing where the water is flowing from I can only offer guesses. If both the hot and cold faucets are off there is also probably a shut off in the shower head also. If all that is off then you possibly have freeze damage such as a cracked line or the faucet itself. You can probably get a replacement faucet at any RV parts store. If it is a cracked line then you have much bigger problems. For a split line cut out a portion of the line and take it to Lowes or Home Depot and they can probably set you up with what it takes to splice in a repair.
Brian, Loretta & Daisy (Golden Retriever)
2008 Holiday Rambler Endeavor 40 PDQ , ISL 400
2014 Ford Explorer toad