There's a few reasons that 'could be'. Try this first. Draw some cold water into a paper cup from the suspect faucet. Go to another sink that hasn't smelled previously and pour the water into the sink drain. Smell it. If it doesn't smell, you most likely have something in the other drain giving you the odor when triggered by the cold water.
Most, but not all campgrounds have a water source that I consider suspect. I've dealt with wells and ponds most of my life as a source of potable water and because of that, I will not ever drink water from a campground. Wells can, and many do, carry bacteria that are fortified by iron and some by sulfur. Then there's the manganese that is also a smell and contaminant. There are filter media that deal with all of these, but I've yet to find one that's portable for campsite use.
Here's an article that may help you find your trouble source:
https://www.mrkitchenfaucets.com/wat...in-one-faucet/