The pair, along with the rest of the film's cast, are now filming Breaking Dawn in Baton Rouge, La., after a stop in Brazil, and have been seen getting cozy around the southern city.
Pattinson, 24 and Stewart, 20 recently joined about two dozen other cast members for a private dinner at Fleming's Prime Steakhouse and Wine Bar, but seemed to only have eyes for one another.
"Kristen and Rob were very couple-like," a restaurant source tells PEOPLE. "There was a lot of hand-holding, sitting next to each other. Everyone was talking around the room, but Kristen and Rob were just talking to each other, not really into the conversations everyone else was having."
At another outing to Cadillac Café, a strip-mall karaoke bar less than a mile from the studio where shooting is taking place, the two were spotted looking "very much a couple," says an onlooker.
"I mean, they weren't slobbering all over each other … but they sure looked like a couple and were talking with each other like a couple," the onlooker says. "They [seem like] a couple in love."
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An employee at Fred's Bar tells PEOPLE that Jonas and Greene have shown up "a couple of times" to listen to bands. "They were definitely together, holding hands," the source says. "A few people recognized them, they were pretty cool."