I think that they do too. I honestly don't know what ships still carry the nuclear Tomahawks. I don't think any of our subs or surface ships are carry these, at least as a manner of course - could be wrong.
Its seems our surface ships do carry nuclear Tomahawks because the Japanese are complaining about it when our ships come to port.
Under a normally defense ready posture commander in chief,who is not trying to save America for the rest of the world to pillage, this information would not be common knowledge. For operational security it is not revealed to anyone outside the upper command structure, and vessel’s exec, CO, and weps team, for obvious reasons- if it’s known to be carrying W class weapons it’s an opportunity for theft or sabotage or detonation.