When you look up a word in a dictionary, let's always pay attention to its word class. Asking about the difference between 不安 and ざわざわ is like asking about the difference between "explosive" and "kaboom". You know they are very different; the former is a noun that works also as an adjective, and the latter is an onomatopoeia (sound of an explosion). They belong to different word classes and thus they are not interchangeable.
Now, 不安 is a noun and a na-adjective. When it works as a noun, it means "anxiety". When it works as a na-adjective, it means "anxious".
ざわざわ is an onomatopoeic adverb and a suru-verb. When it works as an onomatopoeia, it represents a sound — a noise made by people who are confused and/or anxious, like in the picture below. See this article for how onomatopoeic adverbs work in Japanese. When it works as a suru-verb, it means "to buzz (out of anxiety)", "to feel anxiety", "to become uneasy", etc.
In addition, since ざわざわ is an onomatopoeia, it tends to be used to describe a temporary anxiety, confusion, etc. 不安 can be longstanding.