Is "unless it was absolutely necessary" an adverb clause?

No it is not. It is an adverb phrase since it lacks a subordinating conjunction.

Adverb clauses must be preceded or started by a subordinating conjunction such as after, if, because or although. For a sentence to qualify as an adverb clause it must include in its body a subject and a verb.

As a general rule you can use the following formula:

An adverb clause equals a subordinate conjunction plus a subject plus a verb, and it should be answering how? when? where? or why?