Travelers heading to airports this week may face more flight disruptions as air traffic controllers missed their first full paycheck late last month and with the government shutdown on the cusp of becoming the longest in U.S. history.

Almost 6,000 U.S. flights were delayed on Sunday, Nov. 2, according to FlightAware, a flight tracking service. Newark Liberty Airport on Sunday had a four-hour ground delay, while staffing triggers — a Federal Aviation Administration alert system to indicate an air traffic control facility is understaffed — were reported at airports including Nashville, Tennessee; Jacksonville, Florida; Austin, Texas; and Denver, Colorado. 

Airport delays are "going to get worse," Transportation Secretary Sean Duffy said on "Face the Nation with Margaret Brennan" on Sunday. 

The shutdown on Tuesday is set to become the longest in history, when it would eclipse the 2018-2019 funding lapse. In the meantime, air traffic controllers and workers for the Transportation Security Administration — both considered essential workers — are required to show up at their jobs even though their pay is halted until lawmakers agree on a new deal to fund the federal government. 

The longer the shutdown stretches on, the more likely it is to cause flight delays and snarls at security checkpoints, Katy Nastro, a travel expert at Going.com, told CBS News. 

"We're over a month into this shutdown, and in the last few days is really when we've started to see that the seams are ripping," Nastro said. "It's not just in air traffic control staffing triggers, but we're seeing TSA lines hit the 3-4 hour mark."

Some Houston travelers faced waits of about an hour to get through airport security lines this weekend, according to CBS affiliate KHOU. Only two out of five security checkpoints are open at Houston's Bush airport, the station noted, while the Houston Chronicle reported that some people faced waits of three hours to get through the airport's security.

Thanksgiving outlook for air travel

If the shutdown drags longer into November, the real pain point could come during Thanksgiving week, when a record number of Americans are expected to travel for the holidays, Nastro said. 

"I would be cautious if this continues into Thanksgiving — we're likely to see multiple days of 3 million-plus travelers," she said. "At that point, we would be coming into almost two months of a shutdown."

Some holiday travelers might be better off driving to their destinations given the potential for flight disruptions, Nastro added. Booking early morning flights, such as those that depart between 6 a.m. and 7 a.m., can also help minimize delays because these departures are more likely to be on time than flights that are scheduled later in the day, travel data shows. 

Still, experts said that air safety isn't an issue, given that the FAA will slow flights if there aren't enough air traffic controllers directing flights. That was reiterated by Duffy on Sunday, who told "Face the Nation" that the federal government will "stop traffic" if it becomes a safety issue. 

"We're not going to let that happen," Duffy added.