Air travel disruptions will linger into Monday even after a winter storm that forced airlines to cancel more than 14,000 U.S. flights over the weekend clears out.

The winter storm is bringing heavy snow, ice, and sleet, followed by bitter cold across much of the country.

More than 1,870 flights were canceled for Monday, far fewer than over the weekend but a sign that airlines will need a day to recalibrate after the storm. A third of Monday's departures from New York's LaGuardia Airport and John F. Kennedy Airport, and a quarter of Newark Liberty International Airport in New Jersey, were canceled, as were more than half of those out of Boston.

Only 21 Tuesday flights were canceled thus far, however.

Airlines routinely cancel flights ahead of a major storm to prevent planes, passengers and crew from getting stranded, which can worsen disruptions after severe weather clears. January is a low season for travel but the vast storm was snarling travel across the country for hundreds of thousands of people.

American Airlines, Delta Air Lines, JetBlue Airways, Southwest Airlines, United Airlines and other U.S. carriers said they are waiving change and cancellation fees as well as fare differences to rebook for customers with tickets to and from more than 40 airports around the country. The waivers include restrictive basic economy tickets.

American, JetBlue and Southwest are scheduled to report results this week, and their executives are likely to receive questions from analysts about the impact of the storm on their financial results.

The winter storm comes less than a month after airspace was closed over parts of the Caribbean following the U.S. attacks in Venezuela, prompting mass flight cancellations at the end of the holiday season.

On Sunday, airlines canceled 10,251 U.S. flights, according to FlightAware, with affected airports stretching from Texas to Maine.

Nearly 90% of Sunday flights were canceled at LaGuardia Airport in New York and 99% at Ronald Reagan Washington National Airport. More than 74% of flights at Newark Liberty International Airport and John F. Kennedy International Airport, which have more international flights, were also canceled for Sunday.

Fort Worth, Texas-based American Airlines cancelled 1,471 mainline flights on Sunday, close to half of its schedule, while Atlanta-based Delta scrubbed 1,307, or 40%, according to FlightAware. Close to half the flights at Atlanta Hartsfield-Jackson International Airport, the world's busiest, were canceled.

Even airports out of the storm's path were affected as airlines grounded planes.

American said last week that it was repositioning its airplanes and increasing staffing at major airports to help soften the impact of the storm and to "set the stage for a fast, safe recovery once conditions improve."

More than 20 U.S. states were under emergency declarations this weekend due to the massive winter storm.

As of 10:05 a.m. ET Sunday, more than 797,000 U.S. customers were without power, with the worst hit areas being Mississippi, Texas and Louisiana, according to PowerOutage.com.

The Department of Energy issued an emergency order temporarily authorizing grid operator PJM Interconnection to run power generation facilities to help ensure electricity amid extreme temperatures and storm impact across the mid-Atlantic region, though it may exceed environmental permit restrictions.

"As Winter Storm Fern brings extreme cold and dangerous conditions to the Mid-Atlantic, maintaining affordable, reliable, and secure power in the PJM region is non-negotiable," U.S. Energy Secretary Chris Wright said.

The Department of Energy also authorized the Electric Reliability Council of Texas, or ERCOT, to run backup generators for the state.

The National Weather Service warned last week that the winter storm could affect hundreds of millions of people and is expected to "bring widespread heavy snow, sleet, and freezing rain from the Southern Rockies to New England through Monday."

— CNBC's April Roach contributed to this report.