Jessica (came/has come) back from work and she's exhausted.
He (sold/has sold) his car. Someone bought it last week.
Looks like both are correct but I should choose only one.
Both are correct grammar. I assume this is a test question.
Probably they want you to use present perfect in the first (because the second part is present) and past in the second (because the second part is past tense).
The first really does work better with present perfect, there is a clear connection to the present. But the past tense would not be a grammar error.
The second really could be either. Both past and present perfect work fine in the second example, with little or no difference in meaning.