Can using unleaded gas when your car says premium only cause engine shaking and ticking?

Using unleaded fuel should not hurt the car like you mention. It could possibly be a clogged fuel filter or fuel injector. The inexpensive way to check is like you mention, by using a fuel injector cleaner. If that doesn't work, then replace the fuel filter. If neither of those work, then it would be best to let a mechanic check for any other possibilities.