In this case, I'm not talking about ID, but some other new hypothetical theory. How would one go about showing one's theory better explains the evidence than evolution theory?
If you are an ordinary member of the public, who believes they have actual evidence that conflicts with the basis of Evolution (not just some nit-picking claim about a fringe aspect of evolution that is any case a known matter of academic debate), then your first action should probably be to send your data to the Biology dep't of the nearest big university, and ask the specialists there to look at what you are presenting.
If there is any actual value in what you present, then they will probably arrange for you to meet them at the university and discuss plans for further research, perhaps leading to sending a draft paper to an appropriate research journal.
However, it's hard to imagine how any ordinary member of the public is going to discover something like that before the academics do, especially in a field like evolution where hundreds of thousands of professors and post-doctoral research associates have been conducting research for 100 years or more, and on countless aspects that are almost certainly entirely unknown to members of the general public … so it's a million-to-one-on that the research academics would already have known about any such “new” evidence for decades.
If it's a religiously related claim of new evidence undermining evolution, then academics are unlikely to waste their time looking at that, because religious beliefs are not a credible way to determine what is likely to be actually true about anything in the world around us … and, they will already have all seen countless fraudulent ID claims … so they are unlikely to want to waste their time and waste public money on trying to check what anyone presents as a claim of evidence for ID or Creationism.
And as a last simple comment – you may not appreciate how much time & effort typically goes into a piece of science before it gets published as a paper in a real science research journal. But a paper is often the final result of many years of constant daily studies, tests, and experiments … in many aspects of research, a whole research team (maybe 3, 5, or 10 people) might take a decade or more before they have enough high quality data to submit a paper worth publishing (and the Journal will always reject any such paper if the editorial board of international peer-review experts decides that it does not reach the required standard, either for accuracy or as something actually new, novel, and important … many research groups go for many years without being able to get any papers published on what has become their life's work).