Should I ask a new question and explain that it shouldn't be closed as duplicate although it looks like one?
If you can write a question that can also clearly explain why the answers to an existing question don't solve your problem/answer your question, a new question could work. Often, this could be a sign that there is something slightly different between your question and the existing one, and highlighting that part can prevent closure and at the same time show research effort.
If, on the other hand, your question is exactly the same, you're probably better off putting a bounty on the existing question. This can be especially true for questions that were originally answered with a 'workaround'. By adding a bounty (and hopefully attracting a better/different answer) all answers/ways of doing a thing remain in one question, which is better for the site.
Should I down-vote all the answers I don't like?
If an answer doesn't work for you, it isn't useful and you can downvote it. But I would not say you can always just downvote whatever you 'don't like', as that's more of an opinion than an objective metric. Try not to treat downvotes as 'dislikes' and upvotes as 'likes', but vote for the quality of the post.