These sound a lot like Wawas. It's a 24/7 convenience store that also sells sandwiches, soups, salads, coffee, etc. Recently they've been transitioning to a gas station model, but there are stand-alone Wawas too, that are basically the same as the Japanese konbini.
Basically, Wawa just needs to go national already.
Yeah, reading the OP I was like "this sounds a helluva lot like Wawa". For some reason I thought they were more widespread but looking at the store map on Wikipedia it seems like I've just gotten lucky with the places I've lived recently to have them so close by. Royal Farms is/was okay but not on the same level.