Q:

Part of speech: "I am disappointed with"

In a construction such as, "John is disappointed with Alice", what part of speech is disappointed with? It appears to me that the "am" is a linking verb.
Similarly, "Jessica is sad", it seems to me that "sad" is the same part of speech as disappointed with.
One paper I'm reading claims that these are adverbs, but I am pretty sure this isn't the case, as "disappointed" doesn't describe "is". They feel like adjectives, but yet it's not assembled like a typical adjective would be ("Sad Jessica").

A:

People write papers about parts of speech? Good heavens. 
First, yes, be is always an auxiliary verb. Even if it's the only verb in the clause; the lexical item following be in that case is the real predicate.  

(Not a "linking verb", btw; that's grade school stuff, like "5 take away 2")

And disappointed is indeed an adjective -- a predicate adjective since it takes an auxiliary be. It's what's called a "psych predicate", because it refers to a mental state of the subject. Like angry, scared, frightened, mad, surprised, etc.
Since almost all predicate adjectives are intransitive, they can't take objects. However, they can be transitivized with prepositions. The prepositional phrase indicates the stimulus that has caused the mental state to the subject. But the prepositions vary; they're determined by the predicate, as usual.

I'm disappointed. ~ I'm disappointed at Max. ~ I'm disappointed with Max. 
I'm angry. ~ I'm angry at Max. ~ I'm angry with Max. ~ *I'm angry of Max.
I'm mad. ~ I'm mad at Max. ~ *I'm mad with Max. ~ *I'm mad of Max.
I'm scared. ~ *I'm scared at Max. ~ *I'm scared with Max. ~ I'm scared of Max. 
I'm surprised ~ I'm surprised at Max. ~ *I'm surprised with Max. ~ *I'm surprised of Max. 

Psych predicates are also called "Flip predicates" in the literature because they invert the usual roles of the subject as agent and the object as patient (Bill hit Max), instead flipping the subject to patient role, with the prepositional object functioning as cause, if not always agent.
Psych predicates are often formed from past participles (like disappointed and scared) so they have the same form as passive, but they have to be distinguished from them because they don't allow agent by-phrases and they don't refer to an event, but rather a state. I.e, the two sentences below don't mean the same thing.

Bill was scared by Max.  (an event; passive verb)
Bill was scared of Max.  (a mental state; psych predicate adjective)

