An apparatus is provided for a turbine engine. This turbine engine apparatus includes a fuel nozzle. The fuel nozzle includes an airflow inlet, a nozzle orifice, a fuel passage and a swirler passage. The fuel passage is fluidly coupled with the swirler passage through a first fuel aperture in a wall between the fuel passage and the swirler passage. The swirler passage extends along a helical trajectory away from the airflow inlet and towards the nozzle orifice.