Johnny Depp and Amber Heard's Cozy Night Out in L.A.

Amber Heard and Johnny Depp

Madison McGaw/BFAnyc/Sipa USA

By Kirsten Gordon

11/14/2014 AT 07:30 PM EST

Johnny Depp and Amber Heard took a trip to a Los Angeles library this week – but not to read books.

The Hollywood heavyweight and his 28-year-old fiancée were spotted at The Library, a cocktail bar at the Redbury hotel, late on Thursday night. Surrounded by Depp's friends, the pair settled into seating at the back of the book-lined bar in Hollywood for the after-party celebrating the release of Lost on the River, a new album by music supergroup The New Basement Tapes (which features Elvis Costello and Marcus Mumford).

"Amber and Johnny were next to each other all night and seemed very lovey dovey," says a source at the private event, where the couple hung out with the band and enjoyed Signature Library cocktails. "Johnny was talking to the guests and being very approachable."

Earlier, Depp had joined the band on stage at the Montalban Theatre, playing guitar as they performed new interpretations of songs from Bob Dylan's fabled Basement Tapes.

Depp, 51, also makes a cameo in a documentary about the making of the Lost on the River album, which airs on Showtime on Nov. 21.

It wasn't the only L.A. outing this week for the Pirates of the Caribbean star, who got engaged to Heard earlier this year – he's set to be honored at the Hollywood Film Awards on Friday night.