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)
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.