You can run the diesels underwater though. Obviously there is a limit, but surfacing the boat is neither required (sea state permitting) nor desired ("OH LOOK, A SUB!!").
DIW on a CVN is a big deal, that is true (though hardly a career killer by itself for O-gang, even the ones in Reactor Department). NR takes nuclear safety very seriously and the carrier won't sink just from loss of propulsion so the reactors will very much be configured to fail conservative if necessary.
Even on a submarine, which can flounder without propulsion and only has a single reactor, we do scram drills all the time. A scram can occur for no better reason than that reactor protection was feeling finicky that day, so the drills are designed to assume a scram will happen, and recover safely and efficiently.
On the other hand, an unplanned scram really is one of those things which may get someone disqualified from watchstanding, and possibly de-nuked completely, since those usually represent a crew that doesn't properly respect the power of the reactor.