I thought "sarnie" was probably a scouse thing but its origins are uncertain. Some answers I found point to the OED which apparently indicates that the use of "sarnie" popped up in 1961 and is believed to be from a northern England dialectical pronunciation of "sand" from "sandwich."

I was familiar with both this and "butty" from my years in North Wales, I'm not sure about Norfolk where I've lived my last 20.

I think my choice of word usually depends on the filling, eg. a cheese sandwich but a chip (or bacon) butty.