The verb "to get" + particle ...?

Get is a linking verb here, as it means become. So the phrase all crazy is a complement, a phrase that describes the subject of the verb.

I don’t know the term particle phrase, I’m afraid. In the sentence ‘I'm going to get all crazy’, get acts as a copulative verb. The structure is much the same as ‘I'm going to be all crazy.’ 'All crazy' is the complement of 'get', but you may also know it as subject predicative.

It is not a particle phrase. A particle phrase is a phrase consisting of the particle associated with the phrasal verb, the particle's modifiers (if any), and (though some do not include this) the direct object:

He pulled off the sticker. (Or, as some define it: He pulled off the sticker, or He pulled the sticker right off)

In your example, I is the subject, am going to is an idiomatic verb phrase describing the future tense, get is the copula (or linking verb), and all crazy would be called the predicate or subject complement, and specifically a predicate-adjective phrase.

Get has two major semantic uses in English, and any number of idiomatic ones. Semantically, get is an Inchoative (change-of-state) verb; in particular, it is the inchoative form of both be and have.

• He is cold now, but he was not cold before; he got/became cold.
• He has a cold now, but he did not have a cold before; he got a cold.

Become is an inchoative form of be (it's derived from come to be) and therefore can substitute for get when it means come to be, but it can't substitute for get when it means come to have.

• *He became a cold.

Thus, get all crazy is not a phrasal verb, and all crazy therefore can't be a particle phrase. It's just an ordinary predicate adjective phrase (with all as an emphasizer instead of very).

(Parenthetically, this fact is the genesis of I have = I have got = I've got = I got, which we're all familiar with, not to mention hafta = have to = have got to = got to = gotta, and various got ~ gotten usages).