The verbs abjure and adjure both involve solemnity or earnestness, but their meanings are different. To abjure is to recant solemnly, renounce, repudiate, or forswear—for example:
Of course, not all liberals abjure the Constitution. [American Spectator]
Adjure, which is often used in religious contexts,means to appeal or entreat solemnly—for example:
A father-son duo … adjure us to wake up lest America after the cold war repeat the grave errors of Britain after World War I. [NY Times]
If you cannot now bring yourself to threaten veto, then I beg you to strongly adjure the legislature to deliberate long and hard about this abominable legislation. [Chicago Tribune]
In short, when you abjure something, you strongly oppose it. When you adjure someone, you strongly urge them to do something. The direct object of abjure is always the thing you oppose, and the direct object of adjure is always the person whom you urge to act.