can adjectives take gerunds as postmodifiers?

Can adjectives take gerunds or participles as postmodifers? Specifically, I can't figure out how to analyze the following two sentences.

"She is busy working".

"She is happy teaching."

(In each case, it seems like an adjective is taking something other than an adverbial as a modifier. Although "working" and "teaching" could be explained as participial adjectives modifying the subjects, that doesn't really seem to accord with the meaning of the sentences. Likewise, "working" and "teaching" can't be explained as predicate nouns with adjectival modifiers, for obvious reasons.

Since nouns sometimes function as adverbs--for example, "I am going home"; "she arrived Monday"--is it possible that gerunds, which are types of nouns, can likewise function as adverbs in certain situations?)

Similarly, can adjectives take noun clauses as objects in the same way that verbs take noun clauses as objects?

For example, how should the following sentences be scanned?

"I am aware that grammar can be confusing."

(Compare: "I know that grammar can be confusing.")

"I am pleased that you have been promoted"

(Compare: "I heard that you have been promoted")

[Casiopea]

Adjectives can be postmodified by a prepositional phrase, a finite clause, or an infinitive clause. Happy, for example, is an adjective which accepts all three types of postmodifier:

EX: John is happy about the job. (preposition)
EX: John is happy that he got the job. (finite clause)
EX: John is happy to have the job. (infinitive clause)

Source

Looking now at 1) and 2) below, let's try a process of elimination. Since 'working' and 'teaching' are not (a) finite clauses (i.e., they lack tense) or (b) infinitives clause (i.e., '-ing' is not a base form), they must be part of a prepositional phrase, the head of which has been omitted:

1) She is busy (with) working.
2) She is happy (with) teaching.

'with working' and 'with teaching' are prepositional phrases in form. They are made up of the preposition 'with' and its object, the gerunds 'working' and 'teaching'. The entire prepositional phrase postmodifies 'busy' and 'happy', respectively, and it functions as a subject complement, notably a predicate adjective.

Predicate adjective: adjective+prepositional phrase
1) She is busy with working.
2) She is happy with teaching.

Gerunds 'working' and 'teaching' do not directly modify the adjectives 'busy' and 'happy'. They are the objects of the preposition, (with), be it present or not.

1) She is busy working. => Object of omitted (with)
2) She is happy teaching. => Object of omitted (with)

With regards to adverbs, 'enough' is an example of an adverbial postmodifier:

She is happy enough. (Adverb, post modifies 'happy')
She is busy enough. (Adverb, post modifies 'busy')

The same holds true (i.e., omission) for like structures (i.e., adjective + what appears to be a nominal):

I am aware (of the fact) that you have been promoted.
I am pleased (to hear) that you have been promoted.

The underlined portions function as the object of omitted (to hear) and (the fact), respectively.