When you have a new animal that won't acclimate to current conditions (a recently shed, very young water snake that doesn't eat is an animal that is likely failing to feel cozey) you have to start considering what you might change besides food. In other words, the fasting might be a symptom instead of a problem.
You could start by replacing the carpet with a not-too-deep layer of leaves. These will smell more natural (it does make a difference for some specimens) and will provide a more suitable hiding place - everywhere in the tank. The advantage of being able to dive into seclusion on the warm side or the cool side of the tank cannot go without note. Little water snakes hide unless they're hunting or sunning. Uncomfortable snakes won't sun and cool snakes (that can only hide under a chilly water bowl) won't hunt. Leaves also provide superior humidity to a plastic dome over carpet, which might fix the shedding problem.
You didn't mention what color the water bowl is. Get one that's dark on the inside. Sometimes, new specimens can be skittish of bright white (or green or blue or purple depending on the petshop dish) hunting grounds.
Don't handle the animal until it adjusts. New water snakes that "behave" for handling aren't necessarily good snakes. They are more likely scared snakes. Resigning to quiescence is a sign of silent stress just as often as it is a sign of acclimation. If it's stress, the animal won't eat.
You said you had a fifty watt bulb for a basking site. Did you get a thermometer as well? Get a digital thermometer with one of those cord sensors attatched (ten bucks at any large retail store like WalMart or Home Depot) and make sure it tells you that the basking end of the tank is warm enough. In a little ten gallon tank, an air-conditioned room can have a very chilling effect.
Consider putting up an infrared bulb for warmth at night. During the summer months, many water snakes go nocturnal and shun hunting by day. If you turn the little fifty watt energy supply off during the animal's business hours, it will stay, accordingly, closed. Good luck.