NEW YORK (CBSNewYork/AP) — U.S. airlines preemptively canceled thousands of flights Monday and Tuesday as a late-winter storm is expected to dump more than enough snow to disrupt travel in the Northeast.

More than 2,000 flights on Monday and more than 4,100 on Tuesday had been canceled, tracking service said Monday afternoon. Nearly 3,000 flights already had been canceled out of the New York City area as of Monday night.

The Port Authority of New York and New Jersey said there is no plan to close any of the airports officially. But the odds of getting off the ground Tuesday are not in anyone’s favor.

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As CBS2’s Andrea Grymes reported, it was a race against time Monday to get home or out of town before the blizzard hits.

“I said I’m going to get out of here before I get stuck,” said Wanneh Dixon.

Dixon arrived at the Delta Shuttle terminal at LaGuardia – nervous to get home to Washington, D.C. ahead of the blizzard.

“I came in this morning. I should’ve been here three days,” Dixon said. “I decided to get out of here before the snow.”

The shuttle terminal was pretty quiet, but Dixon was not alone.

The impending snowstorm has many travelers scrambling – including Diane and Brian Steele. They got to the American Airlines terminal at LaGuardia hours ahead of time, hoping not to change their evening flight home.

“When we were trying to get an earlier flight, they told us they were all full. Everyone was trying to get out,” Brian Steele said. “We’re just hoping we get off the tarmac.”

Meanwhile, Sheran Keyton of Dallas found herself stranded at LaGuardia before the snowfall even began.

“We’re like no room at the inn here,” Keyton told CBS2’s Jessica Layton.

Keyton’s flight was canceled for Tuesday, and she was having trouble getting any help.

“We spent six hours on the phone trying to get someone,” Keyton said. “We tried on email; the website – it crashed.”

Keyton and her kids raced to the airport to try to get out Monday night. There was no such luck for her, or for many of the others who were packed in the food court with their frustrated faces in their phones.

When asked what he was doing to fill the time, Brian Gibson said, “Trying to find a way out of here.”

But with so many flights already canceled out of the area, that will likely prove impossible until Wednesday at the earliest. And travelers like Trent Harris, who missed his flight home to Arkansas because of the traffic around LaGuardia, were not sure what to do.

“It’s very frustrating. I only have two sets of clothes. We came for a funeral,” Harris said. “Now we’re probably stuck here until Thursday. That’s the soonest they can get us out.”

As travel expert Evan Konwiser explained, the domino effect of disruptions could be felt well into the week. Konwiser said passengers must be proactive starting now.

“Sign up for alerts from the airline or agency,” he said. “It’s up to them to take care of themselves for hotel and meals, and make sure ground transportation are taken care of.”

LaGuardia is also in the midst of a massive construction project, which has not helped traffic concerns recently. But when Chopper 2 flew over Monday afternoon, it was not nearly as bad as we have seen it other times.

As for air traffic, the boards earlier showed hardly any canceled flights. But that was not expected to last.

Chicago bore the largest share of Monday’s cancellations, while Tuesday’s disruptions are hitting hardest from Washington to Boston, including the New York City area.

The major airlines are waiving ticket-change fees that range up to $200 for customers who want to change their travel plans. Restrictions vary by airline.

Southwest has canceled more than 300 flights for Monday and nearly another 900 scheduled for Tuesday, according to FlightAware.

American Airlines, along with its regional partners, canceled more than 530 flights Monday, more than 1,670 flights Tuesday, and 400 flights for Wednesday, the airline said. All Tuesday flights have been canceled at LaGuardia, John F. Kennedy International, and Newark Liberty International airports. The airline is planning to resume normal operations Wednesday at noon.

JetBlue Airways, with major operations in Boston and New York, had already canceled more than 600 flights scheduled for Tuesday, FlightAware said.

(© Copyright 2017 CBS Broadcasting Inc. All Rights Reserved. The Associated Press contributed to this report.)