The singer also uploaded several other photos of the pair with the caption: "Time flies."
Adele also shared a photo of herself holding a fortune cookie that read: "You have found good company – enjoy."
The British singer and dad-of-three Paul first sparked dating rumours when they were spotted sitting together at the NBA Finals in July – just two months after teasing they had been "hanging out".
Adele filed for divorce in 2019 from her husband of one year, Simon Konecki, with whom she shares 9-year-old son Angelo. They had been together since 2011.
The Grammy winner then completely overhauled her image, embarking on a drastic weight loss journey that saw her shed a whopping 45kg.
The exciting personal news comes amid a difficult few months for the singer, who was forced to pull the pin on her much-anticipated Las Vegas residency in January.
Adele took to Instagram at the time with a tearful video saying the show "wasn't ready", just one day before it was due to open.
"I'm so sorry, but my show ain't ready. We've tried absolutely everything that we can to pull it together and have it ready in time for you, but we've been absolutely destroyed by delivery delays and Covid," she began, her voice shaking.
"Half my crew, half my team are down with Covid – still are. It's been impossible to finish the show. I can't give you what I have right now, and I'm gutted," she said.