Reactors may sometimes be shut down at sea to test backup generators and run reactor restarting drills, but it would not be done otherwise. Aircraft carrier propulsion comes from steam from the reactors - if the reactors are offline the ship cannot move. Moreover, the backup generators burn jet fuel, which is convenient because aircraft carriers already have a store of jet fuel for the planes and so don’t have to carry extra fuel for the backup generators, but it’s very expensive, and not something that would be done outside of an emergency or testing emergency preparedness.

Nuclear powered vessels do not spend any significant amount of time shutdown at sea, and there is no reason to. Electrical power is a very small fraction of their total MWh production, with almost all power going to propulsion.

When fuel lifetime becomes an issue for a nuclear naval vessel they will have propulsion limits in place that limit transit speeds to those which are most efficient for the propulsion turbines.