Based on hints given:

The guess “Drag Racing To Jupiter for the big cheese” has two words correct.
The guess “Drag Racing On Taste” has three words correct.
The guess “Race to Taste Moon Cheese” has one word correct.
The guess “Race for A Morsel on Mercury” has one word correct.
The guess “On taste the drag around.” has two words correct.
We also know that the words “Drag” and “On” are correct.

Thus none of the words “Race”, “For”, “A”, “Morsel”, and “Mercury” are correct.
We were told Moon and Cheese are both not correct. So “To” or “Taste” is correct.
Also “Taste” Or “Racing” is correct.
Suppose that “To” is correct. Then because “Taste” is wrong, “Racing” is correct. But then “Drag Racing To” has three correct words.
Suppose that “Taste” is correct. Then “On Taste the Drag Around” has three words correct.
As either “To” or “Taste” is correct. We have a contradiction.

Please explain yourself Arth.