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I think number 1 is an adjective. It describes how she's feeling. Try to substitute another adjective in its place and see if it makes sense. "Sarah is feeling quite pretty now". Pretty obviously is an adjective and that would lead me to believe that "tired" in number 1 is an adjective too.
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dgo is right.
In the first sentence, "tired" is functioning as an adjective. Core sentence: Sarah | is feeling / tired.
Sarah = subject
is feeling = verb
Sentences of the same construction: Sarah is feeling happy, Sarah is feeling angry, Sarah is feeling sick, Sarah is feeling better, and so on. In each case, the linking verb "is feeling" (the verb "to feel" in the present progressive, otherwise known as the present continuous) links the subject with an adjective. The word "tired" has the FORM of a verb, but it is not functioning as a verb in this sentence. Here, it is being used as an adjective. A word like this -- with the form of a verb (-ed, -en, -ing, and others), but functioning in the sentence as an adverb or an adjective -- is known as a participle.
What's the difference in the second sentence? Core sentence: Studying | had tired out | Andrew.
Studying = subject
had tired out = verb (the phrasal verb "to tire out" in the past perfect tense)
Andrew = direct object (receiving the action of the verb)
If the sentence were instead "Andrew is tired" or "Andrew is feeling tired," the word "tired" would be functioning instead as an adjective.
Note that the word "studying" has the FORM of a verb, but it is not functioning as a verb. It is functioning here as a noun. A word like this -- with the form of a verb (always -ing, in this case), but functioning in a sentence as a noun -- is known as a gerund.