Answer:
D: "The days flew by, but the worry never disappeared"
Step-by-step explanation:
Let's go over every answer and see why they're right or wrong.
A: "The days flew by but the worry never disappeared." This sentence is grammatically incorrect. You must put a comma before a conjunction if joining two clauses together.
B: "As the days flew by but the worry never disappeared." When using a preposition, you do not use a conjunction in the same sentence, unless the preposition comes at the beginning of the second clause (for example, "She decided to wait to see if he was okay, but as the days flew by, the worry never disappeared.") Also, as stated before, you must use a comma when using a conjunction to join clauses with a conjunction.
C: "The days flew by and the worry never disappeared." 'And' is a conjunction implying similarity between two clauses, and a person would expect that, as time passed, the worry would lessen at least a little bit, but the sentence subverts that expectation by saying that it never disappeared, implying that no change in the amount of worry had taken place at all. The best conjunction to use would be 'but,' because it would imply contradicting clauses, which is what is happening here.
D: "The days flew by, but the worry never disappeared." This sentence checks all the boxes to be grammatically correct. It uses the correct conjunction, it uses commas correctly, and it doesn't misuse a preposition.