Adverbs Modifying (or Complimenting) Linking Verbs?

Hey, guys. I'm new to the forum, and I'm here to ask tons of questions and drive y'all crazy.

So, let's get the basics out of the way. I know that an adverb describes verbs, adjectives (including predicate adjectives), and other adverbs, but I'm wracking my brain about the N-LV-N sentence pattern with a form of "to be," and an adverb.

Example: "I am definitely a nerd."

Obviously, the adverb "definitely" cannot describe the subject and the predicate noun since they are nouns, so that leaves the linking verb "am" and the article. You can't really modify "am" since it is a state-of-being verb that just links the subject to the predicate noun, so is the adverb just a compliment? I'm wondering where you would diagram it.

This N-LV-A sentence pattern has a totally different sentence structure:

"I am definitely happy."

In this sentence, you can clearly see that the adverb "definitely" is describing the predicate adjective "happy," because it is saying how happy.

And of coarse, we have our prepositional phrases that can work either as an adjective, or adverb phrase: "I am truly the least (of all servants)."

In this N-LV-N sentence, I almost mistook the predicate noun "least" for a predicate adjective, which would totally change how the adverb "truly" and the preposition "of all servants" modifies. But "of all servants" is working as an adjective phrase that modifies the pronoun "least," and again, we are left with an adverb and the linking verb "am."

So what does the adverb really do?