One important reason to keep a shamash candle burning in a hanukkiah is because Hanukkah candles are not supposed to be used except for fufilling the mitzvah. The shamash candle allows one to use its light for other purposes like reading or finding ones way in a dark room.
This works for a while after the candles are lit, but some hanukkiot (menorahs) have a shamash holder that causes that candle to go out sooner than the others or a shamash could burn faster than the other candles just from the random variation in wax volume or flame size. In this case, when a shamash is burned down and out, I think you should not use remaining candles for any mundane purpose.
What practical steps should one take if they find themselves in this situation? I can imagine:
Blowing out the other candles if the minimum thirty minutes have passed and it is not Shabbat.
Replacing that original shamash with another shamash.
Turning on the electric lights in the room so that the remaining candles are only used for the mitzvah.
What have you read or been taught that is relevant to this situation? What would be acceptable ways to handle it? Are there other ways that I have not thought of?