An apparatus is provided for a turbine engine. This apparatus includes a fuel-air mixer, and the fuel-air mixer includes an inner passage, a sidewall, a steam passage, a fuel nozzle and an air swirler. The inner passage extends axially along an axis within the fuel-air mixer. The sidewall extends circumferentially around and axially along the inner passage. The steam passage is embedded within the sidewall and extends along the inner passage. The fuel nozzle is configured to direct fuel into the inner passage. The air swirler is configured to direct swirled air into the inner passage for mixing with the fuel.