Enlisted members swear the same oath.
That's because it was the enlisted oath, the officer's oath has not changed since 1868, and is the same oath that everyone but the President takes, including the VP, the cabinet officers and all federal public officials. The POTUS oath is specified by the Constitution. That oath says nothing about obeying orders.
The confusion may have arisen, because most officers enlist first, if only in the reserves, in the case of ROTC cadets, and thus that that oath, before their commissioning oath. National Guard officers take a similar but slightly different oath.
However neither the 1950 or the 1962 revisions to the oath of enlistment speak of *lawful* orders. But their was some emphasis on that aspect in some training I vagley remember taking, or maybe conducting, when I was an Air Guardsman, in the mid 1980s.