And to answer the question in your title, no, you cannot run away to another state to be emancipated. First - NY does NOT have a statute for minors to be emancipated - at best a minor who is living on his own WITH permission AND who is supporting him or herself may be perceived as emancipated. What you have read is not correct.

Second, NO state will emancipate a minor who is not a resident of their state, particularly if they ran away to get there.