"oral exam" sounds funny. I can't help but imagine a person (whom I visualize as Bryan Cranston for obvious reason) examining someone with their mouth or having their mouth examined :-]
That happens more often than you might think, plus it makes everything consistent.
One of the mean songs for Verbal is “Of, relating to, or associated with words.” So a written test could be a verbal test.
Then there's 'interrogation', which Heisenberg might have concurred with...
First definition at MW:
> of, relating to, or consisting of words
> When two or more parties come to an agreement without any written documentation, they create a verbal agreement (known formally as an oral contract).
I think that alone demonstrates that verbal contrasts with written.