I clean my carrots with a plastic scrubby for cleaning pots and pans and hold it under running water as I scrub it. I keep it separate from my cleaning scrubbies. My mom used to peel them with a potato peeler to clean them.
I use a vegetable scrub cloth and hydrogen peroxide.
I just use a vegetable brush under running water. The pets and I go through 3+ pounds of carrots a week, lot of carrot cleaning going on here.
Peggy....I wash then use the peeler for double safety.
And then there is this: