Actually, the governor does not appoint congressmen, only Senators.
I think you are wrong. If, for instance, a Congressman dies early on in his term, the state DOES select an interim replacement (usually his wife). The mechanism may vary by state (legislator or Governor), and I don't recall the specifics for Louisiana (I've been living out of state for the last 5-6 years), but I am certain that a mechanism for such replacement does exist.