(CNN) — Last week’s winter weather travel mess is lingering like a vicious hangover into this week — and the headaches are migraine-proportioned for Southwest Airlines, its CEO Bob Jordan, airline employees and most of all its frustrated passengers on Monday.

But Southwest accounts for a whopping share of those. None of the other US carriers have canceled nearly as many flights or as much of their schedule as Southwest.

The Dallas-based airline had canceled more than two-thirds of its flights — close to 2,800 total — as of 7 p.m. Monday, according to FlightAware.

At one point, it canceled around 300 flights in the span of a half hour Monday afternoon.

And it looks like the Southwest pain will spread into Tuesday. More than 1,600 of those flights were already canceled as of 7 p.m. ET Monday. That’s the lion’s share of the 1,760 flight cancellations reported so far for Tuesday for all US airlines.

CEO: ‘A lot of issues in the operation’

On Christmas night, ahead of Monday’s meltdown, Jordan told employees the airline has “a lot of issues in the operation right now.”

CNN was provided a transcript of the message to Southwest employees by an aviation source.

Jordan told employees, “Part of what we’re suffering is a lack of tools. We’ve talked an awful lot about modernizing the operation, and the need to do that.

On social media, customers are complaining loudly about long lines to speak with representatives, problems with lost bags and excessive wait times or busy signals on the airline’s customer service telephone lines. One passenger told CNN her family was on the phone for 10 hours with Southwest.

CNN’s Carlos Suarez spoke with frustrated passengers in line at the Southwest ticket counter at Hartsfield-Jackson Atlanta International Airport on Monday.

He’s reportedthat about 150 customers were waiting in a long line at one point to rebook, with the line snaking around the back of the ticket counter.

‘Disruptions across our network’

Southwest responded to the massive cancellations in an emailed statement Monday afternoon:

“With consecutive days of extreme winter weather across our network behind us, continuing challenges are impacting our Customers and Employees in a significant way that is unacceptable,” the statement read.

“We’re working with Safety at the forefront to urgently address wide-scale disruption. … On the other side of this, we’ll work to make things right for those we’ve let down, including our Employees.”

In an earlier statement to CNN on Monday, Southwest Airlines said it is “experiencing disruptions across our network as a result of (the winter storm’s) lingering effects on the totality of our operation.”

Some of the airports seeing the biggest issues are Denver, Las Vegas, Chicago Midway, Baltimore/Washington, Dallas Love Field and Phoenix Sky Harbor.

Customers encountered long lines at Southwest counters on Monday at Raleigh-Durham International Airport.

WRAL

Calls made Monday afternoon by CNN to Southwest’s customer service did not go through, so customers couldn’t even get in the queue to speak to a representative. Southwest told CNN it is “fully staffed to answer calls.”

The airline also says, “those whose flights have been canceled may request a full refund or receive a flight credit, which does not expire.”

One of the replies in part read: “Stop blaming the WEATHER! Had to buy a first class ticket on another airline but it TOOK OFF ON TIME! You still have our luggage with medication inside! Can’t get through on the phone!”

Ripple effects at Southwest

In the Sunday night message to employees, Jordan said, “We need to be able to produce solutions faster.” He said the airline is “committed to and invested in” improving its systems.

The president of the union that represents Southwest’s flight attendants told CNN’s Pamela Brown in an on-air interview Monday that those systems have left its members stranded.

“The phone system the company uses is just not working. They’re just not manned with enough manpower in order to give the scheduling changes to flight attendants, and that’s created a ripple effect that is creating chaos throughout the nation,” Lyn Montgomery, president of TWU Local 556, told CNN.

She said some flight attendants needed to sleep in airports as a result.

Flight crew treatment called ‘despicable’

Montgomery expressed outrage on behalf of workers and customers late Monday afternoon in a news release.

“The way Southwest Airlines has treated its flight crews can only be termed ‘despicable,'” said Montgomery. “We know the demands of holiday travel. We know winter storms. And believe me, we know about stepping up and putting in long work hours when we are called to do so; we are flight attendants.

“But at this point, the many years of failure by management, despite many unions’ demands to modernize, has left flight attendants fatigued, stranded, hungry and cold — on Christmas! This impacts lives and threatens safety for all.”

“The company’s failures are happening year-round, but are particularly egregious on Christmas,” Montgomery said. “Our customers struggled with it just as our thousands of flight attendants did.”

Kathleen Bangs, a FlightAware spokesperson, said in on-air interview that Southwest’s schedule includes shorter flights with tighter turnaround times that are causing some of the problems. “Those turnaround times bog things down,” Bangs said.

CNN has reached out to the airline for comment on Jordan’s video message.

What can stranded passengers do?

“The main hotline for US airlines will be clogged with other passengers getting rebooked. To get through to an agent quickly, call any one of the airline’s dozens of international offices” Scott Keyes said.

“Agents can handle your reservation just like US-based ones can, but there’s virtually no wait to get through.”

A 10-hour phone call

Michelle Perkins, her husband JJ and their six children had their flights canceled while waiting at the airport in Las Vegas on Christmas Eve. The family was headed to Orlando for a surprise holiday trip for the kids, she said.

Perkins was traveling with kids from ages 17 years old to 15 months. She told CNN that when their flight was canceled, they headed to baggage claim to get their luggage along with their car seats.

“We went to baggage to retrieve our luggage and car seats, waited over an hour with six kids in tears only to be told that … bags and car seats must continue on to their destination in Orlando.”

Perkins said her husband was on a phone call with Southwest Airlines for nearly 10 hours to get their flights refunded and file a claim on their baggage. She added that they were told their luggage was at the Las Vegas airport.

“We went back to the airport to pick it up, and they denied that our luggage was still there, we told them that it was tracked at Las Vegas, so then they changed their story and said it was in a secure location on the tarmac and they cannot get it because there are too many bags.”

As Monday afternoon, they have yet to be reunited with their checked luggage.

Perkins said they are just thankful that they are all together for the holiday and were able to go back home to Las Vegas instead of being stranded in another city.

Any relief in sight?

It could be next week before this is all completely sorted out.

“When there’s more than 10,000 flight cancellations over the past week, it takes time for airlines to work through and reaccommodate the backlog of travelers,” Keyes told CNN Travel in an email.

“While it will depend on the weather forecast (which looks promising for much of the country) and what number of travelers wind up canceling their holiday plans, I’d expect that by next week, things will have largely returned to normal,” Keyes said.

And why are so many people having trouble rebooking?

“One complicating factor for people hoping to get reaccommodated is the fact that there are so few available seats this season,” Keyes said.

“That’s both because Christmas and New Year is one of the most popular times of year for travel, and because the number of flights on the schedule this year is still down 15-20%, making the challenge even steeper for those needing to get rebooked.”

Buffalo still reeling

Meanwhile, in hard-hit western New York, Buffalo International Airport said in its most recent tweet that it plans to resume passenger flights at 11 a.m. ET Tuesday.

The temperature at the airport was 19 degrees Fahrenheit (minus 7 Celsius) around 7 p.m. ET Monday, with light snow falling on top of the huge amounts the area has already seen.

And it’s not just the airport affected there. Road travel remained treacherous because of extreme wintry conditions.

“The City of Buffalo is impassable in most areas, while mains may have a lane open for emergency traffic or two, most secondaries as well as side streets have not been touched yet,” Poloncarz said.

He adding that the cleared main roads are primarily for the use of life-saving measures to open up areas around hospitals and nursing homes.

A rough past week

A winter storm that swept across the US was ill-timed for travelers who had started pushing Christmas week flying numbers back toward pre-pandemic levels.

On Christmas Day, there were 3,178 flights canceled and 6,870 flight delayed, according to FlightAware.

On Christmas Eve, there were a total of 3,487 flights canceled, according to FlightAware.

Friday was the worst day of this streak with 5,934 cancellations, while Thursday saw almost 2,700 cancellations.

This megablast of winter weather across the eastern two-thirds of the nation is forecast to slowly moderate this week.

CNN’s Ross Levitt, Amanda Jackson, Rebekah Reiss, Chris Boyette and Artemis Moshtaghian contributed to this story.