Yvonne Young bought her first snowshovel ever this month, to clear the walk outside her District Heights town house and to dig her 1978 Granada out of snowdrifts. She also has invested more than $100 on new sweaters, thermal underwear, heavy pants and woolen leggings for her sons John, 15, and Alvin, 5. Those unexpected expenses, said the 33-year-old postal worker, have helped make this "a winter I'd like to forget."

Her $20,000 salary usually enables her to break even, Young said yesterday as she stood outside the McBride's store off Minnesota Avenue NE. But since the cold wave and heavy snows began here Jan. 7, she said, "I'm starting to fall behind . . . and wait 'til I get my January gas heating bill. That's gonna be a killer."

The numbing cold, averaging 15 degrees below normal in the past three weeks, and the heavy, lingering snow have combined to wreak havoc with many family budgets that already were stretched thin from Christmas buying. In a region unaccustomed to intense winters, the higher heating bills, emergency car and home repairs, and additional spending on clothing and even food have added a bit more misery to the miserable weather.

On the other hand, some businesses, ranging from clothiers that sell cold-weather gear to travel agencies that sell warm-weather fantasies, have found profit in the cold and snow.

Washington Gas Light Co. reported yesterday that it had 102,318 delinquent accounts as of Dec. 31--more than 18 percent of all customers, with overdue bills totaling $8,365,000. Both figures are apparently all-time records for the utility. "Obviously this is a reflection of hard times," said spokesman Paul Young. Gas prices are up 19 percent over January 1981 and the utility is pumping about 4 percent more gas than last January, he said.

The District of Columbia Energy Office this month is receiving more than 500 calls per day, mostly from low-income people falling further behind on fuel bills, according to a spokesman.

While utilities and fuel oil dealers will benefit from cold temperatures, many businesses in the metropolitan area have suffered a tremendous slackening of trade because of snowstorms and snarled traffic, according to Greater Washington Board of Trade spokesman Leonard Kolodny. "Even if people want to buy warmer merchandise . . . or snow tires, chains, or shovels, they can't get there," he said.

Restaurants and movie theatres likewise have been devastated by weather, said Gus Ladas, president of the Restaurant Association of Metropolitan Washington, who said sales at his three restaurants are down 20 percent to 50 percent from normal January levels.

Some businesses--those able to thrive on winter--are doing fine, due to the frenzied buying of snow- and cold-crazed consumers.

The Safeway supermarket chain not only has sold out of snow shovels throughout the metropolitan area, according to spokesman Ernie Moore, but also has had higher-than-normal food sales whenever weather predictions turn dire. "People really stock up whenever there is a scare," Moore said. "Business has been excellent. The lines were incredible" at some stores.

Yvonne Young said her two boys seem to burn more energy and need more food these days. In addition, Prince George's County schools have been closed three days and the boys stay home and play, then eat, then play and eat some more, she said.

Her car is eating more gasoline, too, Young said, because she has frequently been mired in slow-moving or nonmoving traffic between District Heights and her job at the main post office on Capitol Hill. She estimated she has spent $30 to $40 more on gasoline this month than usual.

"It's good for business," said Tom Dooley, the automotive supplies manager at Montgomery Ward's Capital Plaza store in Hyattsville. "It's the best winter in many years." He estimated automotive sales at 15 percent above normal.

As the weather has worsened, motorists seem to have suddenly realized their needs and descended on his department in droves, Dooley said. "First week of January, it was anti-freeze . . . This past week it was windshield solvent. We sold out of solvent on Friday." Now, he said, customers are stocking up on a solution to keep their gas-lines from freezing.

His department also sold out of about 150 sets of tire-chains last week, Dooley said. Some of the chains, which sell for $9.99 to $28 a pair, had been sitting on shelves for about five years, he said.

Those who fail to get their cars in shape have helped make January a bonanza for auto-towing services. The Potomac branch of the American Automobile Association had logged 35,000 emergency road-service calls by Monday, with calls continuing at about 2,000 a day, a spokesman said.

The Sears department store chain has had its "biggest month ever" in sales of furnaces and hot-water heaters at the 23 stores in the region, according to merchandise manager Richard Demert, who said sales on water-pipe insulation and kerosene space heaters also "went crazy" since the cold wave.

Some clothiers too have benefitted. "We've had close-to-incredible volume," said Doug Braswell, manager of Eddie Bauer Inc., outdoor outfitters who specialize in down-insulated clothing. January's cold has made the month almost as profitable as pre-Christmas season, quite a rarity for a retailer, he said.

"People have been clamoring for boots, wool gloves, knit hats and long underwear," he said. During the salvage operations following the Air Florida crash, he said, television networks spent hundreds of dollars at the store outfitting camera crews warmly enough to stand outdoors for hours.

Hot-selling items, he said, are down-insulated parkas (from $79 to $359); long woolen underwear ($14 to $15 each for tops and bottoms); boots with rubber bottoms, leather tops and wool lining ($40 to $60); wool caps ($5 to $7) and even down-insulated face masks ($24.95.)

Among the other beneficiaries of cold weather are those businesses that thrive on fantasies of warmth. The Here Today There Tomorrow travel service, for instance, has had an upsurge of business this month, much of it from people depressed by the weather and making plans for the Caribbean, said office manager Marcia Selva.

Cold weather also helped the Anthony Pool company, which has contracted to build about 25 in-ground pools this month at costs of $10,000 and up, according to Fairfax office manager Nancy Hogan. After slow sales in November and December, the pool firm is getting customers who are sick of winter by now and begin thinking of summertime, she said.

Last week, Hogan said, a woman in a heavy winter coat arrived, saying, "You probably think I am crazy, but I am here to discuss a pool."