On Friday, the singer, 38, was seen out and about for the first time since news of her divorce filing was made public. Clarkson was photographed taking her dog for a walk around her neighborhood while wearing a pair of sunglasses.
“Kelly is spending time in L.A. with her kids,” a source tells PEOPLE of the star, who shares daughter River, 6, and son Remington, 4, with the music manager, 43, who is also dad to daughter Savannah, 18, and 13-year-old son Seth.
“She has been low-key and mostly stayed at home,” the source adds of Clarkson. “She seems to be doing okay.”
Citing irreconcilable differences, The Voice coach filed for divorce in L.A. on June 4, according to documents obtained by PEOPLE. The papers list the couple's separation date as "TBD."
Ahead of the launch of her syndicated talk show, The Kelly Clarkson Show, she opened up about their busy lives in a PEOPLE cover story last year after they decided to move their family to L.A. from Nashville permanently to be closer to Clarkson's work.
"We ended up moving our whole family out here. It was kind of a big decision for us, but it just seemed like a new adventure, honestly,” Clarkson told PEOPLE last August. “My husband is a really good salesman on ideas. He's just really good at pointing out things that maybe will push me out of my comfort zone, which is why we're really great partners in life as well."
Also in the interview, Clarkson discussed their “really busy” schedule. "Even the show here, he has his own office. I have my own office. And he's always on the phone ... he's always working,” she said of Blackstock, a longtime manager for Clarkson, as well as other A-listers including Blake Shelton.
"When we vacation — we just took our first family vacation and had a really good chunk of time because we knew things were about to be really busy. He goes off and does stuff with the kids, and then I'll do something else with the other two. We do a really good job of being like, 'I need some me time,'" she added.
Since March, Clarkson and her family have been social distancing in Montana.
All had seemed well between the couple in recent months, as Clarkson documented their rustic life on their ranch. Blackstock even recruited their kids to record a sweet video to wish Clarkson a happy birthday back in April: "Happy birthday, baby. I love you more than you'll ever know," he said in the clip.