Jennifer Lopez Beams at Alex Rodriguez as Couple Cozies Up in NYC 1 Week After Engagement

Big Apple date night!

Jennifer Lopez and Alex Rodriguez were seen stepping out for dinner in New York City on Saturday, just one week after the retired Yankees star popped the question during their romantic Bahamas getaway.

During their night out, Lopez, 49, was photographed casting a loving look towards Rodriguez, 43, as the pair walked arm-in-arm to Italian restaurant Carbone.

Lopez wore a chic black velvet suit with subtle sparkles, which she paired with a matching bag and a tucked-in blue shirt. Opting for a casual ensemble, Rodriguez paired a black leather jacket with a black t-shirt, jeans and a pair of black boots.

Earlier on Saturday, Rodriguez shared that he couldn’t be happier “to be back in the greatest city in the world.”

“There’s nothing like walking around New York,” he said in an Instagram video, before giving his former baseball team a shout-out. “You can smell opening day right around the corner. Yankee baseball. Best fans, best franchise, best.”

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New York New York ????

A post shared by Alex Rodriguez (@arod) on Mar 16, 2019 at 6:41pm PDT

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A source close to Lopez recently told PEOPLE that the vacation proposal was the “best surprise ever.”

“Jennifer can’t stop looking at her ring,” the source said. “She is obsessed and loves it. She keeps saying that the proposal was the best surprise ever.”

“They were going for a beach walk when Alex asked her to marry him. Jennifer loves that the proposal was so romantic and low-key,” the insider revealed. “She loves that Alex knows her so well and felt comfortable keeping it real instead of going for a flashy proposal. It was a dream proposal for Jennifer. She wouldn’t have wanted it any other way.”

Alex Rodriguez and Jennifer Lopez
Alex Rodriguez and Jennifer Lopez

The pair first announced their engagement on Instagram last week and waited a few days before sharing intimate photos from the romantic beachside proposal.

In one image, the retired athlete gets down on one knee on the beach, popping the question at sunset to a surprised Lopez.

“Locking it down. #futuremrsrodriguez,” the athlete captioned his Instagram post on Tuesday. Lopez also shared four more photos, including one in which the newly engaged couple share a passionate post-proposal kiss.

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Rodriguez opened up about the experience on his Instagram Story on Tuesday, writing, “There aren’t many words to share with you, only feelings. Gratitude. Excitement. And just … joy. We know there is work to do to keep building a family and future we can be proud of, and we can’t do it without all of you. We are so thankful for all your messages of love and congratulations.”

While they aren’t in wedding planning mode yet, a source previously told PEOPLE their kids will “definitely” play a huge role in their upcoming wedding.

Lopez and ex-husband Marc Anthony, 50, have 11-year-old twins Maximilian “Max” David and Emme Maribel, while Rodriguez and ex-wife Cynthia Scurtis, 46, share daughters Ella, 10, and Natasha, 14.

Jennifer Lopez, Alex Rodriguez and kids
Jennifer Lopez, Alex Rodriguez and kids

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“For them both, they are still all about their family with the four kids. Their initial strong attraction for each other would have quickly faded if they had not been able to come together as a family,” a source told PEOPLE.

“Their kids all accepted each other and got along from the beginning. It’s always tricky when you merge families, but Jennifer and Alex did a great job from the start. The kids are amazing together. And they have wanted Jennifer and Alex to get married for a long time. They will all definitely be a huge part of the wedding,” the source added.