My experiment supports a theory. Shall I contact the theorist attaching with my drafts? We did an experiment and the results unexpectedly support a famous theory. I am not entirely sure how to write the full paper or sell the results because I am not too familiar with that theory. And I don't know which journals welcome that theory. I think chatting with that theorist will surely help.
Shall I directly send our results and drafts to that professor, or shall I send a short cover letter first, which concisely describes our results and shows how his theory is supported, and then send the paper only if the theorist agrees?

I talked to my advisor and they agree I could contact this person, but I'm not sure how to start. The theory is famous for its philosophical and mathematical elegancy but one of its important implications is hard to test.
This is a complicated topic and you should definitely ask your advisor. There could be all sorts of potential pitfalls, e.g. perhaps there are caveats/loopholes to your experiment, perhaps you only supported a restricted form of the theory, perhaps the theory had unknown parameters that your experiment helped fix, and so on.
Given the dangers, don't try to navigate this yourself. If it comes to contacting the theorist, your advisor is likely to know more about what to write, as well.