Answer:
1. Our answers are correct, but their answers are incorrect.
2. Your clothes are new and your shoes are new.
Explanation:
In number 1, we have two independent clauses with contrasting ideas. The first one affirms the subject's answers are correct. The second one affirms the other subject's answers are incorrect. For that reason, it is best to use the conjunction "but", which conveys a contrasting idea:
Our answers are correct, but their answers are incorrect.
In number two, the two clauses are expressing similar ideas instead of contrasting ones. The second idea basically adds to the first one: someone has new things - clothes and shoes. The best conjunction in this case is "and", which conveys an idea of addition:
Your clothes are new and your shoes are new.
