You don’t need to turn it off at the curb cock, just in the house. There should be a main valve in the basement at floor level, turn that off, then drain the system in the house by opening up any taps you have in the basement, if none there, then the lowest in the main floor.
If any standing water remaining in pipes happens to freeze, it shouldn’t cause damage as there is expansion room with all the air in the pipes.
The city will charge you to turn the water off at the street, and make you wait around for them to arrive. Don’t go that way unless you plan to do maintenance like replace your main shut off valve.