My experiment supports a theory. Shall I contact the theorist attaching with my drafts?

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.

My advice would be to send a note introducing yourself and explaining what you have done and the results. Don't "flood" the person in the initial contact. It is too easy to assume such mails are from cranks. I guess mathematicians still get letters with "squaring the circle" constructions.

Offer to share further if you like. Offer to collaborate if you like. Say you have some questions if you think you need their help.

But a shorter note is more likely, I think, to lead to more.