He starts out in the movie wearing that light-colored coat with the sheriff's badge plainly visible. But throughout the rest of the movie, whenever he wears that coat, there's no sheriff's badge visible.
Remember in "The Three Godfathers" when "Pearl" Ward Bond is talking to the cowboys in front of his home and no one can see his sheriff badge until he decides to let them know who the real "Pearl" is. Eric it might be an overlooked prop or it may have been a deliberate act to show that the Marshall or Sheriff is not always on duty. You have a good eye for detail, that type of focus leads to much more enjoyment of the film industry.
A few things. First he's the Sheriff of a small town. Everyone knows who he is. Secondly, he's been targeted by Burdette and has a price on his head. Why advertise to would be assassins? And as others have pointed out he's not always on duty.
I thought that was very good attention to detail by the filmmakers. It's a pity they forgot to tell John Wayne to take his wedding ring off. :)
I don't think he is playing a widower, no! Although of course ultimately that's up to the viewer to decide.
Dean Martin wore quite a bit of jewellery in real life. It's very noticeable how he would always remove it when playing roles where it would have been anachronistic. It's something that's impressed me in the past, particularly when so many people are quick to denounce him as "not a real actor". He clearly knew how to pay attention to detail though, when a lot of his peers didn't. Check out his first Western, "Pardners". Jerry Lewis wears an astonishing amount of jewellery in that film, and none of it looks right in a period piece. Maybe sentimental attachment prevents these people from removing such things? It's a shame, though. Once you've noticed it, you can't easily ignore it again.
Jack Webb ALWAYS wore a bracelet given to him by his first wife, Julie London. He wears it even playing the hard-as-nails drill instructor in THE D.I. or the gruff night editor in -30- and of course, as the just-the-facts Joe Friday. "May I bone your kipper, Mademoiselle?"