PG-
Your understanding is correct.
Steam cools as it expands.
If the steam remains under pressure the steam exhaust will be hotter than when the STG is under full load and the steam pressure is "consumed" or used up by the turbine.
If you are just learning about the nature of gases, you can learn more about this by studying the laws of partial pressures, and the ideal gas laws.
google "partial pressure of gases" and "ideal gas law" to read about the relationship of temperature, pressure, and volume.