Yes, we always use the Skywest callsign. You can tell if it's Delta Connection or United Express flight by the flight number. All DC flights are 3XXX numbers and all UEX flights are 6XXX numbers. Our flights are handled by our own Skywest dispatchers in St. George, UT.
There have been times when some pilots, more like flight attendants that will mess up and say Skywest/Delta Connection on a United Express flight. . . of course I've never done that