Can adjectives be placed in front of verbs, e.g. “The duck was busy diving for food”?

The positional nature of English grammar gives your two sentences different structures in spite of the similarity of their wording. To see this, let's look at a similar pair of sentences that have the same structure:

1a. The busy duck was diving for food.
2a. The duck was diving busily for food.

(Ignore the slightly awkward placement of "busily" in the second sentence.)

The basic structure of both sentences is

Subject - Verb - Adverbial Prepositional Phrase

Subject = "duck"
Verb (past progressive tense) = "was diving"
Prepositional Phrase (of purpose) = "for food"

The difference, of course, is that the first sentence has at attributive adjective ("busy") modifying "duck," and the second has an adverb of manner ("busily") modifying "was diving." The different modifiers, however, don't affect the basic structure.

But things are different for the original first sentence:

1. The duck was busy diving for food.

Now the structure of the sentence is

Subject - Copulative Verb - Predicate Adjective - [Present Participle Phrase]

Subject = "the duck"
Copulative Verb (simple past tense) = "was"
Predicate Adjective = "busy"
[Present Participle Phrase] = "diving for food"

The Present Participle Phrase is in brackets because I'm delaying describing it's function in the sentence, but in any case, the participle "diving" isn't part of the verb as in sentence 1a.

So what is it? You may decide that it's an adverbial phrase of purpose modifying the verb telling us why the duck was what it was, or you may decide that it's a nominative absolute, associated with but independent of the subject-verb combination.