Schiff, along with everyone else, had to swear an oath to tell the truth and only the truth. But he doesn’t care about oaths that interfere with his lying.
Unfortunately, the House Managers did not take an oath to tell the truth at all. They and the Senators took an oath to conduct themselves in accordance with the Constitution. Since Article 1, section 6 of the Constitution gives cover to members of the Legisalture for anything they say in either the House or the Senate, there is no recourse when they lie there.
I don't know what kind of oath the defense attorneys took. They may have had to swear to tell the truth, the whole truth and nothing but the truth.
The framers of the Constitution know that there would be some bad members of Congress but probably did not dream how the entire majority would collude in deceitful behavior.