What kind of evidence can you offer to demonstrate that you are right and I am wrong?

Who says I need "evidence" to argue that you should do something? I could rely on my perceived authority - in fact, you could take this as a definition of what "moral authority" is all about. Sometimes that moral authority comes from religion (or cosmology, more generally), sometimes it's derived from tradition, etc. So I have to dispute your claim that:

it's pretty much in the nature of a moral claim to be part of a debate or discussion.

since it is quite self-evident that many people and institutions have made moral claims in the past that were not perceived as propely being part of a "debate" or "discussion". It's true that, sometimes, moral claims are seen in such a way - especially when they're seen as originating from individual instinct and cognition, and thus leading people to think of themselves as being on the "right side" of an ethical dilemma or conflict. And yet, at some level, more formalized systems like law and politics presumably rely on widespread trust in the "system" itself as a moral authority, if only one with a very limited scope.

So, you're never going to get an answer to the question of "what a normative claim is", because the whole concept involves a kind of tension. There's an "authority to be followed" side, and an "internal moral cognition" side, and both can be right to some degree and even interact in a fruitful way.